Off-diagonal Elements Research Articles

Open quantum systems with Kadanoff-Baym equations

We study the temporal evolution of quantum mechanical fermionic particles exhibiting one bound state within a one-dimensional attractive square-well potential in a heat bath of bosonic particles. For this open quantum system we formulate the non-equilibrium Kadanoff-Baym equations for the system particles by taking the interactions to be elastic 2-2 scatterings with the heat-bath particles. The corresponding spatially inhomogeneous integro-differential equations for the one-particle Greens's function are solved numerically. We demonstrate how the system particles equilibrate and thermalize with the heat bath and how the off-diagonal elements of the density matrix, expressed in the one-particle energy eigenbasis, decohere, so that only the diagonal entries, i.e. the occupation numbers, survive. In addition, the time evolution of the (retarded) Green's function also determines the spectral properties of the various one-particle quantum states.

Branch dependence of halfwidths: Theoretical analysis of N2-broadened halfwidths of CH3Br in the v6 band

The N2-broadened halfwidths of CH3Br in the ν6 band have been calculated, along with the relaxation matrices W. These calculations employ our modified and refined versions of the Robert-Bonamy formalism. Extensive comparisons between the predicted halfwidths from the models and experimental measurements at T = 296 K are presented. Our latest model very closely matches the measurements. Meanwhile, we considered line mixing effects in the spectral profile of the RQ(j,0) sub-branch. Our method probably yields too large values of the off-diagonal elements of W. Finally, by comparing our calculated halfwidths with the data in HITRAN, we have observed generally good agreement, especially for lines with high values of j. However, the agreement becomes poor for lines with low values of j, deteriorating as j decreases. We identified a factor contributing to this poor agreement: the neglect of branch dependence in the development of HITRAN data. We believe that for lines with low values of j, this oversight undermines the reliability of the halfwidth values of CH3Br in HITRAN.

Generation of Scale-Free Assortative Networks via Newman Rewiring for Simulation of Diffusion Phenomena

By collecting and expanding several numerical recipes developed in previous work, we implement an object-oriented Python code, based on the networkX library, for the realization of the configuration model and Newman rewiring. The software can be applied to any kind of network and “target” correlations, but it is tested with focus on scale-free networks and assortative correlations. In order to generate the degree sequence we use the method of “random hubs”, which gives networks with minimal fluctuations. For the assortative rewiring we use the simple Vazquez-Weigt matrix as a test in the case of random networks; since it does not appear to be effective in the case of scale-free networks, we subsequently turn to another recipe which generates matrices with decreasing off-diagonal elements. The rewiring procedure is also important at the theoretical level, in order to test which types of statistically acceptable correlations can actually be realized in concrete networks. From the point of view of applications, its main use is in the construction of correlated networks for the solution of dynamical or diffusion processes through an analysis of the evolution of single nodes, i.e., beyond the Heterogeneous Mean Field approximation. As an example, we report on an application to the Bass diffusion model, with calculations of the time tmax of the diffusion peak. The same networks can additionally be exported in environments for agent-based simulations like NetLogo.

Asymmetric-coupled Ge/SiGe quantum wells for second harmonic generation at 7.1 THz in integrated waveguides: a theoretical study

Abstract We present a theoretical investigation of guided second harmonic generation at THz frequencies in SiGe waveguides embedding n-type Ge/SiGe asymmetric coupled quantum wells to engineer a giant second order nonlinear susceptibility. A characteristic of the chosen material system is the existence of large off-diagonal elements in the χ 2 tensor, coupling optical modes with different polarization. To account for this effect, we generalize the coupled-mode theory, proposing a theoretical model suitable for concurrently resolving every second harmonic generation interaction among guide-sustained modes, regardless of which χ 2 tensor elements it originates from. Furthermore, we exploit the presence of off-diagonal χ 2 elements and the peculiarity of the SiGe material system to develop a simple and novel approach to achieve perfect phase matching without requiring any fabrication process. For a realistic design of the quantum heterostructure we estimate second order nonlinear susceptibility peak values of ∼7 and ∼1.4 × 105 pm/V for diagonal and off diagonal χ 2 elements, respectively. Embedding such heterostructure in Ge-rich SiGe waveguides of thicknesses of the order of 10–15 μm leads to second harmonic generation efficiencies comprised between 0.2 and 2 %, depending on the choice of device parameters. As a case study, we focus on the technologically relevant frequency of 7.1 THz, yet the results we report may be extended to the whole 5–20 THz range.

An Inverse Spectral Problem for Non-Self-Adjoint Jacobi Matrices

Abstract We consider the class of bounded symmetric Jacobi matrices $J$ with positive off-diagonal elements and complex diagonal elements. With each matrix $J$ from this class, we associate the spectral data, which consists of a pair $(\nu ,\psi )$. Here $\nu $ is the spectral measure of $|J|=\sqrt {J^{*}J}$ and $\psi $ is a phase function on the real line satisfying $|\psi |\leq 1$ almost everywhere with respect to the measure $\nu $. Our main result is that the map from $J$ to the pair $(\nu ,\psi )$ is a bijection between our class of Jacobi matrices and the set of all spectral data.

Diagrammatic multiplet sum method (MSM) density functional theory (DFT): Investigation of the transferability of integrals in "simple" DFT-based approaches to multideterminantal problems.

Kohn-Sham density functional theory (DFT) typically works well for describing dynamic correlation. Two other types of correlation, arising in the cases of degenerate (static) or quasidegenerate (nondynamic) zero-order states, represent a difficult problem for DFT. When symmetry is present, multiplet sum method (MSM) DFT [Ziegler et al., Theor. Chim. Acta 4, 877 (1977)] provides one of the earliest and simplest ways to include static correlation in DFT. MSM-DFT assumes that DFT provides a good description of single-determinant energies and uses symmetry and simple ansätze to include the effects of static correlation. This is equivalent to determining the off-diagonal matrix elements in a small configuration interaction (CI) eigenvalue problem. Our ultimate goal, however, is nondynamic correlation in cases where symmetry is inadequate for fixing the dynamic-correlation limitation of DFT. To this end, we have developed a diagrammatic approach to MSM-DFT, which does not, by itself, solve the nondynamic correlation problem in DFT but which facilitates comparison with wave function CI and so allows educated guesses of off-diagonal CI matrix elements even in the absence of symmetry. In every case, an additional exchange-only ansatz (EXAN) allows the MSM-DFT formulas to be transformed into wave function formulas. This EXAN also works for transforming time-dependent DFT into time-dependent Hartree-Fock. Although not enough to uniquely guess DFT formulas from wave function formulas, the diagrammatic approach and the EXAN provide important constraints on any guesses that might be used. We illustrate how diagrammatic MSM-DFT may be used to guess a nondynamic correlation correction for the dissociation of H2 and how diagrammatic MSM-DFT may be used to guess a nonsymmetry-based coupling element in the O2 multiplet problem, which is reasonably close to a previous symmetry-derived result.

Elastic full-waveform inversion in tilted transverse isotropy media using Gauss Newton optimization

In anisotropic full waveform inversion (FWI), the underground medium is commonly assumed to be vertical transverse isotropy (VTI). However, in reality, the presence of stress and tectonic movements often leads to a nonzero tilt angle θt of the strata's symmetry axis. This introduces variations in the direction of the fast axis during the propagation of seismic waves. Therefore, it is more practical and beneficial to consider tilted transverse isotropy (TTI) media when dealing with complex geological structures, as it enhances the accuracy of imaging and inversion. However, the consideration of TTI media also introduces a significant challenge of parameter coupling, commonly known as crosstalk or trade-off. This refers to the situation where the update of one parameter affects and projects onto other parameters. To address this issue and improve the inversion accuracy, we incorporate the Hessian within the inversion process. The off-diagonal elements of the Hessian matrix specifically provide important information about the correlations between different parameters, which aids in their decoupling and mitigates the effect of crosstalk. In numerical experiments, we applied the Gauss-Newton (GN) method to TTI elastic FWI and compared the results with those obtained using the preconditioned conjugate gradient (P-CG) method.The inversion results indicate that the GN method can better alleviate the parameter coupling, making simultaneous inversion of multiple parameters possible. θt

Coherence as an indicator to discern electromagnetically induced transparency and Autler–Townes splitting

Electromagnetically induced transparency (EIT) and Autler–Townes splitting (ATS) are generally characterized and distinguished by the width of the transparency created in the absorption profile of a weak probe in presence of a strong control field. This often leads to ambiguities, as both phenomena yield similar spectroscopic signatures. However, an objective method based on Akaike’s information criterion (AIC) test offers a quantitative way to discern the two regimes when applied on the probe absorption profile. The obtained transition value of control field strength was found to be higher than the value given by pole analysis of the corresponding off-diagonal density matrix element. By contrast, we apply the test on ground state coherence and the measured coherence quantifier, which yields a distinct transition point around the predicted value even in presence of noise. Our test accurately captures the transition between two regimes, indicating that a proper measure of coherence is essential for making such distinctions.

A Decoupling Control Method for Converting DIDO Sequence Admittance Model of Grid-Tied Converter System Into SISO

This paper proposes a decoupling control method to eliminate frequency coupling in three-phase grid-tied system. Firstly, the admittance model of three-phase <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> -type three-phase grid-tied converter (GTC) considering DC-bus voltage control (DVC) loop and phase-locked loop (PLL) is established in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> -domain frame, and the influence of asymmetrical DVC and PLL control structure on frequency coupling oscillation is analyzed. Then decoupling factors are added to the DVC loop and current control (CC) loop to achieve that the off-diagonal elements of the sequence admittance matrix of the GTC are 0, which means the frequency coupling in the GTC is eliminated. Therefore, the GTC system can be regarded as a single input single output (SISO) system, and the stability can be analyzed by the classic Nyquist criterion in the sequence-domain frame. Compared with the double input double output (DIDO) system, the resulted SISO system convenient for calculating the stability margin of the system and guiding parameter design. In addition, the SISO system is easier to be analyzed on the oscillation and resonance mechanism, and the analysis results can be used to optimize the design of the converter controller, grid planning and operation. Finally, simulation and experiment verify the feasibility and effectiveness of the proposed decoupling control method.

Zombie cats on the quantum-classical frontier: Wigner-Moyal and semiclassical limit dynamics of quantum coherence in molecules.

In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

Optimization of Weight Matrices for the Linear Quadratic Regulator Problem Using Algebraic Closed-Form Solutions

This work proposes an analytical gradient-based optimization approach to determine the optimal weight matrices that make the state and control input at the final time close to zero for the linear quadratic regulator problem. Most existing methodologies focused on regulating the diagonal elements using only bio-inspired approaches or analytical approaches. The method proposed, contrarily, optimizes both diagonal and off-diagonal matrix elements based on the gradient. Moreover, by introducing a new variable composed of the steady-state and time-varying terms for the Riccati matrix and using the coordinate transformation for the state, one develops algebraic equationsbased closed-form solutions to generate the required states and numerical partial derivatives for an optimization strategy that does not require the computationally intensive numerical integration process. The authors test the algorithm with one- and two-degrees-of-freedom linear plant models, and it yields the weight matrices that successfully satisfy the pre-defined requirement, which is the norm of the augmented states less than 10−5. The results suggest the broad applicability of the proposed algorithm in science and engineering.

NISQ algorithm for the matrix elements of a generic observable

The calculation of off-diagonal matrix elements has various applications in fields such as nuclear physics and quantum chemistry. In this paper, we present a noisy intermediate scale quantum algorithm for estimating the diagonal and off-diagonal matrix elements of a generic observable in the energy eigenbasis of a given Hamiltonian without explicitly preparing its eigenstates. By means of numerical simulations we show that this approach finds many of the matrix elements for the one and two qubits cases. Specifically, while in the first case, one can initialize the ansatz parameters over a broad interval, in the latter the optimization landscape can significantly slow down the speed of convergence and one should therefore be careful to restrict the initialization to a smaller range of parameters.

High-frequency transport properties of a 2D electron gas with spin-orbit interaction under magnetic field driven topological transition

We study the transport properties of a two-dimensional electron system with an essential spin-orbit interaction under topological phase transition due to changing in-plane magnetic field. We have analyzed the contribution of diagonal and off-diagonal components of density matrix with respect to number of spin zone in the high-frequency conductivity. Analytical formulas for the conductivity tensor have been obtained. A numerical analysis giving full enough representation about the dependence of the conductivity on frequency and magnetic field is presented. We have shown that the contribution of the off-diagonal elements of density matrix is the greater than the higher the frequency of the alternating electromagnetic field. It has been established that for magnetic field value at which the minimum electron energy of upper spin zone exceeds the Fermi energy, the conductivity as a function of magnetic field change in discrete step.

Electroweak baryogenesis via top-charm mixing

We investigate a scenario of electroweak baryogenesis in the two Higgs doublet model with quark flavor mixing. In general, off-diagonal components of quark Yukawa interactions with additional Higgs bosons are strongly constrained by the data for flavor changing neutral currents. However, top-charm quark mixing is not the case, so that a large off-diagonal element can be taken, which can contribute to generating baryon asymmetry of the universe. It turns out that CP violating phases of the off-diagonal element in the source term in the Boltzmann equation are eliminated by the rephasing. This result is somewhat different from previous works on electroweak baryogenesis by flavor off-diagonal Yukawa couplings. Instead, we find that the absolute value of the top-charm off-diagonal element enhances CP violating phases in the Higgs potential, by which sufficient amount of the baryon number can be generated to explain the observed baryon asymmetry of the universe. We find that such a scenario is viable under the current experimental data. The model can be tested by the current and future measurements of various flavor experiments like Kaon rare decays, in addition to high energy collider experiments as well as gravitational wave observations. Characteristic predictions of our model would be deviations in Kaon rare decays. Branching ratios of K+→π+νν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {K}^{+}\ o {\\pi}^{+}\ u \\overline{\ u} $$\\end{document} and KL→π0νν¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {K}_L\ o {\\pi}^0\ u \\overline{\ u} $$\\end{document} can deviate by the order of 10% and 1%, respectively, which may be tested at future Kaon experiments such as NA62 and KOTO step-2.

Dependence in constrained Bayesian optimization

AbstractConstrained Bayesian optimization optimizes a black-box objective function subject to black-box constraints. For simplicity, most existing works assume that multiple constraints are independent. To ask, when and how does dependence between constraints help?, we remove this assumption and implement probability of feasibility with dependence (Dep-PoF) by applying multiple output Gaussian processes (MOGPs) as surrogate models and using expectation propagation to approximate the probabilities. We compare Dep-PoF and the independent version PoF. We propose two new acquisition functions incorporating Dep-PoF and test them on synthetic and practical benchmarks. Our results are largely negative: incorporating dependence between the constraints does not help much. Empirically, incorporating dependence between constraints may be useful if: (i) the solution is on the boundary of the feasible region(s) or (ii) the feasible set is very small. When these conditions are satisfied, the predictive covariance matrix from the MOGP may be poorly approximated by a diagonal matrix and the off-diagonal matrix elements may become important. Dep-PoF may apply to settings where (i) the constraints and their dependence are totally unknown and (ii) experiments are so expensive that any slightly better Bayesian optimization procedure is preferred. But, in most cases, Dep-PoF is indistinguishable from PoF.

Stabilization of the Solution to the Initial-Boundary Value Problem for One-Dimensional Isothermal Equations of Viscous Compressible Multicomponent Media

The paper considers the initial-boundary value problem for one-dimensional isothermal equations of viscous compressible multicomponent media, commonly known as a generalization of the Navier — Stokes equations. The studied equations include higher derivatives of the velocities of all components, unlike the Navier — Stokes equations, where viscosity is a scalar variable. Viscosity forms a matrix of elements responsible for viscous friction due to the composite structure of viscous stress tensors for the multicomponent case. Diagonal elements of the matrix stand for viscous friction within each component, and off-diagonal elements stand for friction between components. Such complication does not allow the automatic extension of the known results for the Navier — Stokes equations to the multicomponent case. Thus, for the diagonal matrix, the equations would be linked only through the lower terms. The paper considers a more complex case of an off-diagonal viscosity matrix. We prove the stabilization of the solution of the initial-boundary value problem with an unlimited increase in time without simplifying assumptions about the structure of the viscosity matrix, except for the standard physical requirements of symmetry and positive definiteness.

Mathematical model to calculate the total number of decays in Peptide Receptor Radionuclide Therapy using nonlinear mixed effect modelling

The calculation of an individual total number of decays or time-integrated activities (TIAs) of a radiopharmaceutical for kidneys is desirable for dosimetry in molecular radiotherapy. The accuracy of the TIAs calculation relies heavily on the chosen fit model. Therefore, this study aimed to determine the best mathematical model of 177Lu-DOTATATE in peptide receptor radionuclide therapy (PRRT) using the nonlinear mixed effect (NLME) and model selection method. Pharmacokinetics data of 177Lu-DOTATATE in the kidneys of ten PRRT patients were obtained from the literature (PMID: 33443063). Eleven sums of exponential (SOE) functions were fitted to the pharmacokinetics data in the NLME framework. The model selection was performed based on the goodness of fit test and the corrected Akaike Information Criterion (AICc). The goodness of fit was evaluated based on the fitted graphs visualization, coefficient of variations (CV<50%), and the off-diagonal elements of the correlation matrix (-0,8≤CM≤0,8). In general, all SOE functions were successfully fitted to the pharmacokinetic data of 177Lu-DOTATATE in kidneys. The function was selected as the best mathematical model with an AICc weight of 77.58 %.

Optimal reconciliation of hierarchical wind power forecasts of correlated wind farms

Wind power forecasts of widely distributed wind farms in a spatial hierarchy are required at different hierarchical levels of power system architecture for various operational decisions. Accurate hierarchical forecasts necessitate aggregation consistency across hierarchy i.e. lower-level forecasts summate precisely to higher-level forecasts. Aggregation consistency substantially improves through optimal reconciliation of forecasts. Reconciliation can further improve forecast accuracy by incorporating spatial correlation among wind farms in the hierarchy. However, covariance matrix formed in reconciliation observes matrix order to increase manifold while including correlation. Estimation of such high-dimensional matrix is computationally complex. Additionally, aggregation of estimation errors leads to poor estimation accuracy. Moreover, in this reconciliation process, covariance matrix based on the observed data is non-invertible as the number of predictors usually happens to be larger than sample size. This work proposes MinT(Shrink) estimator for covariance matrix estimation for such systems. This estimator captures correlation among wind farms while shrinks off-diagonal elements of covariance matrix toward zero to alleviate computational complexity of high-dimensional matrix. It also provides invertible covariance matrix by employing shrinkage parameter λ. Case study presents efficacy of considering spatial correlation in hierarchical wind power forecasts in terms of improved forecast accuracy at all hierarchy levels and time horizons.

The Effect of Spatially Correlated Errors on Sea Surface Height Retrieval from SWOT Altimetry

The upcoming technology of wide-swath altimetry from space will enable monitoring the ocean surface at 4–5 times better spatial resolution and 2–3 times better accuracy than traditional nadir altimeters. This development will provide a chance to directly observe submesoscale sea surface height (SSH) variations that have a typical magnitude of a few centimeters. Taking full advantage of this opportunity requires correct treatment of the correlated SSH errors caused by uncertainties in environmental conditions beneath the satellite and in the geometry and orientation of the on-board interferometer. These observation errors are highly correlated both along and across the surface swath scanned by the satellite, and this presents a significant challenge for accurate processing. In particular, the SWOT precision matrix has off-diagonal elements that are too numerous to allow standard approaches to remain tractable. In this study, we explore the utility of a block-diagonal approximation to the SWOT precision matrix in order to reconstruct SSH variability in the region east of Greenland. An extensive set of 2dVar assimilation experiments demonstrates that the sparse approximation proposed for the precision matrix provides accurate SSH retrievals when the background-to-observation error ratio ν does not exceed 3 and significant wave height is below 2.5 m. We also quantify the range of ν and significant wave heights over which the retrieval accuracy of the exact spatially correlated SWOT error model will outperform the uncorrelated model. In particular, the estimated range is found to be substantially wider (ν&lt;10 with significant wave heights below 8–10 m), indicating the potential benefits of further improving the accuracy of approximations for the SWOT precision matrix.

Machine learning classification can significantly reduce the cost of calculating the Hamiltonian matrix in CI calculations.

Hamiltonian matrices in electronic and nuclear contexts are highly computation intensive to calculate, mainly due to the cost for the potential matrix. Typically, these matrices contain many off-diagonal elements that are orders of magnitude smaller than diagonal elements. We illustrate that here for vibrational H-matrices of H2O, C2H3 (vinyl), and C2H5NO2 (glycine) using full-dimensional abinitio-based potential surfaces. We then show that many of these small elements can be replaced by zero with small errors of the resulting full set of eigenvalues, depending on the threshold value for this replacement. As a result of this empirical evidence, we investigate three machine learning approaches to predict the zero elements. This is shown to be successful for these H-matrices after training on a small set of calculated elements. For H-matrices of vinyl and glycine, of order 15 552 and 8828, respectively, training on a percent or so of elements is sufficient to obtain all eigenvalues with a mean absolute error of roughly 2 cm-1.