Off-diagonal Elements Research Articles

Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time.

The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.

Epitaxial growth and polarized Raman scattering of niobium dioxide films

We report the structural, electrical, and optical characterization of epitaxial niobium dioxide (NbO2) films fabricated on MgF2(001) substrates. The films were almost stoichiometric, had an indirect bandgap of 0.7 eV, and exhibited a phase transition at ∼1080 K. A polarized Raman scattering study of the films was conducted to investigate the Raman symmetry in the low-temperature phase. Based on the angular-dependent polarized Raman spectra, we assigned 13 modes to Ag symmetry and 14 to Bg symmetry. We also evaluated the Raman tensor elements of the Bg modes and found that the off-diagonal elements were nearly zero in most of the Bg modes, except for a phonon mode at 267 cm−1. This study aids understanding of the lattice dynamics of NbO2, which plays a critical role in the phase transition.

Precision millimetre-wave spectroscopy and calculation of the Stark manifolds in high Rydberg states of para-H[formula omitted]

Precision measurements of transitions between singlet (S=0) Rydberg states of H2 belonging to series converging on the X+2Σg+(v+=0,N+=0) state of H2+ have been carried out by millimetre-wave spectroscopy under field-free conditions and in the presence of weak static electric fields. The Stark effect mixes states with different values of the orbital-angular-momentum quantum number ℓ and leads to quadratic Stark shifts of low-ℓ states and to linear Stark shifts of the nearly degenerate manifold of high-ℓ states. Transitions to the Stark manifold were observed for the principal numbers 50 and 70, at fields below 50 mV/cm, with linewidths below 500 kHz. The energy-level structure was calculated using a matrix-diagonalisation approach, in which the zero-field positions of the ℓ≤3 Rydberg states were obtained either from multichannel-quantum-defect-theory calculations or experiment, and those of the ℓ≥4 Rydberg states from a long-range core-polarisation model. This approach offers the advantage of including rovibronic channel interactions through the MQDT treatment while retaining the advantages of a spherical basis for the determination of the off-diagonal elements of the Stark operator. Comparison of experimental and calculated transition frequencies enabled the quantitative description of the Stark manifolds, with residuals typically below 50 kHz. We demonstrate how the procedure leads to quantum defects and binding energies of high Rydberg states with unprecedented accuracy, opening up new prospects for the determination of ionisation energies in molecules.

Development of a double-MRT pseudopotential model for tridimensional boiling simulation

A new thermal lattice Boltzmann scheme for the simulation of heat transfer and phase change in multiphase flows based on the pseudopotential formulation is introduced. The model includes a novel elaboration of the equilibrium distribution function, source term, and off-diagonal elements of the relaxation matrix. The resulting equation formally recovers the macroscopic advection-diffusion equation for energy transport avoiding unwanted non-physical terms. Moreover, the thermal diffusivity can be controlled using relaxation factors and free parameters of the equilibrium distribution. The new model is applied to simulate the stratification of a van der Waals fluid and the one-dimensional Stefan problem. In particular, the predictive capability of the model is tested against real experimental conditions, finding good agreement in the bubble growth rate and the wall-temperature dependence of the departure diameter.

Unrestricted, restricted, and regularized models for forecasting multivariate volatility

Abstract We perform an extensive investigation of different specifications of the BEKK-type multivariate volatility models for a moderate number of assets, focusing on how the degree of parametrization affects forecasting performance. Because the unrestricted specification may be too generously parameterized, often one imposes restrictions on coefficient matrices constraining them to have a diagonal or even scalar structure. We frame all three model variations (full, diagonal, scalar) as special cases of a ridge-type regularized estimator, where the off-diagonal elements are shrunk towards zero and the diagonal elements are shrunk towards homogeneity. Our forecasting experiments with BEKK-type Conditional Autoregressive Wishart model for realized volatility confirm the superiority of the more parsimonious scalar and diagonal model variations. Even though sometimes a moderate degree of regularization of the diagonal and off-diagonal parameters may be beneficial for forecasting performance, it does not regularly lead to tangible performance improvements irrespective of how precise is tuning of regularization intensity. Additionally, our results highlight the crucial importance of frequent model re-estimation in improving the forecast precision, and, perhaps paradoxically, a slight advantage of shorter estimation windows compared to longer windows.

Many-Body Perturbation Theory with Localized Orbitals: Accounting for Localization Diagrams as Integral Dressing.

We argue that the so-called localization diagrams, originating from off-diagonal Fockian elements, do not have to be dealt with explicitly in the Davidson–Kapuy many-body perturbation theory with localized orbitals but can be accounted for by dressed two-electron integrals.

An efficient GPU-parallel coordinate descent algorithm for sparse precision matrix estimation via scaled lasso

The sparse precision matrix plays an essential role in the Gaussian graphical model since a zero off-diagonal element indicates conditional independence of the corresponding two variables given others. In the Gaussian graphical model, many methods have been proposed, and their theoretical properties are given as well. Among these, the sparse precision matrix estimation via scaled lasso (SPMESL) has an attractive feature in which the penalty level is automatically set to achieve the optimal convergence rate under the sparsity and invertibility conditions. Conversely, other methods need to be used in searching for the optimal tuning parameter. Despite such an advantage, the SPMESL has not been widely used due to its expensive computational cost. In this paper, we develop a GPU-parallel coordinate descent (CD) algorithm for the SPMESL and numerically show that the proposed algorithm is much faster than the least angle regression (LARS) tailored to the SPMESL. Several comprehensive numerical studies are conducted to investigate the scalability of the proposed algorithm and the estimation performance of the SPMESL. The results show that the SPMESL has the lowest false discovery rate for all cases and the best performance in the case where the level of the sparsity of the columns is high.

Effect of Quantum Coherence on Landauer's Principle.

Landauer’s principle provides a fundamental lower bound for energy dissipation occurring with information erasure in the quantum regime. While most studies have related the entropy reduction incorporated with the erasure to the lower bound (entropic bound), recent efforts have also provided another lower bound associated with the thermal fluctuation of the dissipated energy (thermodynamic bound). The coexistence of the two bounds has stimulated comparative studies of their properties; however, these studies were performed for systems where the time-evolution of diagonal (population) and off-diagonal (coherence) elements of the density matrix are decoupled. In this paper, we aimed to broaden the comparative study to include the influence of quantum coherence induced by the tilted system–reservoir interaction direction. By examining their dependence on the initial state of the information-bearing system, we find that the following properties of the bounds are generically held regardless of whether the influence of the coherence is present or not: the entropic bound serves as the tighter bound for a sufficiently mixed initial state, while the thermodynamic bound is tighter when the purity of the initial state is sufficiently high. The exception is the case where the system dynamics involve only phase relaxation; in this case, the two bounds coincide when the initial coherence is zero; otherwise, the thermodynamic bound serves the tighter bound. We also find the quantum information erasure inevitably accompanies constant energy dissipation caused by the creation of system–reservoir correlation, which may cause an additional source of energetic cost for the erasure.

Identification of a state of persistent spin helix in a parallel magnetic field, and exploration of its transport properties

The tensors of conductivity and spin susceptibility of a two-dimensional electron gas (2DEG) with equal Rashba and Dresselhaus spin-orbit interaction constants [the persistent spin helix (PSH) state] in a parallel magnetic field are calculated. Applying a parallel magnetic field to the PSH state leads to the appearance of a saddle point in the spectrum and, accordingly, to a Van Hove singularity (VHS), whose amplitude increases indefinitely as the magnetic field strength $(\mathbf{b})$ approaches to some critical value ${b}_{cr}$. The presence of a VHS in the density of states is an important factor determining the conductivity and spin susceptibility tensors. When only the lower spin subband is filled, the off-diagonal elements of the conductivity tensor are nonzero; that is, a Hall voltage arises due to the anisotropy of the Fermi surface and scattering. In the region where two spin subbands are filled, the diagonal and off-diagonal components of the spin susceptibility tensor are equal, and the off-diagonal terms of the conductivity tensor vanish. A sharp decrease in the conductivity and spin susceptibility at the beginning of the filling of the second spin subband, the ratio of the diagonal component of the spin susceptibility to the off-diagonal one, and also the number of critical points in the spectrum make it possible to establish the PSH 2DEG state.

Performance of the collective three-level quantum thermal engine

We investigate the performance of a microscopic quantum heat engine consisting of V- or Lambda-type emitters interacting collectively or independently when being in contact with environmental thermal reservoirs. Though the efficiency of a Carnot's cycle is always higher than those associated with these setups, we have found that the performance of the cooperative Lambda-type heat engine may be larger than that of the V-type under similar conditions. Cooperativity among the emitters plays an important role for the Lambda-type setup, significantly improving its performance, while is less relevant for a V-type thermal engine. This is because the population inversion on the working atomic transition as well as its off-diagonal elements behave differently for these two atomic ensembles.

Observation of optical gyromagnetic properties in a magneto-plasmonic metamaterial

Metamaterials with artificial optical properties have attracted significant research interest. In particular, artificial magnetic resonances with non-unity permeability tensor at optical frequencies in metamaterials have been reported. However, only non-unity diagonal elements of the permeability tensor have been demonstrated to date. A gyromagnetic permeability tensor with non-zero off-diagonal elements has not been observed at the optical frequencies. Here we report the observation of gyromagnetic properties in the near-infrared wavelength range in a magneto-plasmonic metamaterial. The non-zero off-diagonal permeability tensor element causes the transverse magneto-optical Kerr effect under s-polarized incidence that otherwise vanishes if the permeability tensor is not gyromagnetic. By retrieving the permeability tensor elements from reflection, transmission, and transverse magneto-optical Kerr effect spectra, we show that the effective off-diagonal permeability tensor elements reach 10−3 level at the resonance wavelength (~900 nm) of the split-ring resonators, which is at least two orders of magnitude higher than magneto-optical materials at the same wavelength. The artificial gyromagnetic permeability is attributed to the change in the local electric field direction modulated by the split-ring resonators. Our study demonstrates the possibility of engineering the permeability and permittivity tensors in metamaterials at arbitrary frequencies, thereby promising a variety of applications of next-generation nonreciprocal photonic devices, magneto-plasmonic sensors, and active metamaterials.

Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics.

We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state. New flux densities are introduced where in both, the electronic and the nuclear case, the main contribution is contained in the component corresponding to the BO state. As a consequence, they can be determined within the BO approximation, and an excellent agreement with the exact results is found. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Nuclear data uncertainty in iterative neutron spectrum unfolding

ABSTRACT We developed an algorithm for propagating the uncertainty of dosimetry cross sections to that of the neutron spectrum obtained through multiple-foil activation and unfolding process. The algorithm derives the uncertainty of the solution spectrum as the standard deviation of a set of solution spectra obtained by unfolding with numerous sets of dosimetry cross sections, which are randomly generated from the evaluated cross sections and their covariances in a data library. We simulated a multiple-foil activation experiment in a C(d,n) neutron field by PHITS code for some dosimetry reactions to test the algorithm. The GRAVEL code analyzed production amounts to unfold neutron spectrum and then our new algorithm derived the uncertainties of the resultant neutron spectrum originated in the nuclear data uncertainty. Further, we found the uncertainty of the solution spectrum becomes smaller if we ignore the off-diagonal elements of the covariance matrices of the dosimetry cross sections.

Proton-neutron pairing and binding energies of nuclei close to the N=Z line

We analyze the contribution of isovector and isoscalar proton-neutron pairing to the binding energies of even-even nuclei with $N\ensuremath{-}Z=0, 2, 4$ and atomic mass $20<A<100$. The binding energies are calculated in the mean-field approach by coupling a Skyrme-type functional to an isovector-isoscalar pairing force of zero range. The latter is treated in the framework of quartet condensation model (QCM), which conserves exactly the particle number and the isospin. The interdependence of pairing and deformation is taken into account by performing self-consistent Skyrme-HF $+$ QCM calculations in the intrinsic system. It is shown that the binding energies are not changing much when the isoscalar pairing is switched on. This fact is related to the off-diagonal matrix elements of the pairing force, which are less attractive for the isoscalar force, and to the competition between the isoscalar and isovector pairing channels.

Interacting bosons in a triple well: Preface of many-body quantum chaos.

Systems of interacting bosons in triple-well potentials are of significant theoretical and experimental interest. They are explored in contexts that range from quantum phase transitions and quantum dynamics to semiclassical analysis. Here, we systematically investigate the onset of quantum chaos in a triple-well model that moves away from integrability as its potential gets tilted. Even in its deepest chaotic regime, the system presents features reminiscent of integrability. Our studies are based on level spacing distribution and spectral form factor, structure of the eigenstates, and diagonal and off-diagonal elements of observables in relationship to the eigenstate thermalization hypothesis. With only three sites, the system's eigenstates are at the brink of becoming fully chaotic, so they do not yet exhibit Gaussian distributions, which resonates with the results for the observables.

Theoretical study of electronic and optical properties in doped quantum structures with Razavy confining potential: effects of external fields

We investigate the energy states of confined electrons in doped quantum structures with Razavy-like confining potentials. The theoretical investigation is performed within the effective mass and parabolic band approximations, including the influence of externally applied electric and magnetic fields. First, we analyze the case of a Razavy quantum well and determine its conduction subband spectrum, focusing on the lowest energy levels and their probability densities. These properties have been numerically determined by self-consistently solving the coupled system of Schrödinger, Poisson, and charge neutrality equations. Doping is introduced via an on-center $$\delta$$ -like layer. In order to evaluate the associated total (linear plus nonlinear) optical absorption coefficient (TOAC), we have calculated the corresponding diagonal and off-diagonal electric dipole matrix elements, the main energy separation, and the occupancy ratio which are the main factors governing the variation in this optical response. A detailed discussion is given about the influence of doping concentration as well as electric and magnetic fields, which can produce shifts in the light absorption signal, toward either lower or higher frequencies. As an extension of the self-consistent method to a two-dimensional problem, the energy states of quantum wire system of circular cross section, with internal doping and Razavy potential, have been calculated. The response of eigenvalues, self-consistent potentials and electron densities is studied with the variation in $$\delta$$ -doping layer width and of the donor density. Finally, the origin of Friedel-like oscillations, that arise in the density profile, generated by the occupation of internal and surface electronic states has been explained.

Anderson localization and multifractal spectrum at the transition point in a two-dimensional non-Hermitian AII† system

The Anderson localization transition in a two-dimensional AII† system is studied by eigenvalue statistics and then confirmed by the multifractal analysis of the wave functions at the transition point. The system is modeled by a two-dimensional lattice structure with real-quaternion off-diagonal elements and complex on-site energies, whose real and imaginary parts are two independent random variables. Via finite-size scaling analysis of eigenvalue spacing ratios, we find the non-Hermiticity reduces the critical disorder and give an estimate of the critical exponent ν = 1.89, showing the system belongs to a new universal class other than the AII class and probably shares the same exponent with two-dimensional Hermitian DIII systems although they have different symmetries. The Anderson localization transition is further confirmed by checking the linearity in the parametric representation of the singularity strength and by checking the universality of the forms of the singularity spectra of different system sizes. The generalized dimensions are obtained as and .

Clustering with general photo-z uncertainties: application to Baryon Acoustic Oscillations

ABSTRACT Photometric data can be analysed using the 3D correlation function ξp to extract cosmological information via e.g. measurement of the Baryon Acoustic Oscillations (BAO). Previous studies modeled ξp assuming a Gaussian photo-z approximation. In this work we improve the modeling by incorporating realistic photo-z distribution. We show that the position of the BAO scale in ξp is determined by the photo-z distribution and the Jacobian of the transformation. The latter diverges at the transverse scale of the separation s⊥, and it explains why ξp traces the underlying correlation function at s⊥, rather than s, when the photo-z uncertainty σz/(1+ z) ≳ 0.02. We also obtain the Gaussian covariance for ξp. Due to photo-z mixing, the covariance of ξp shows strong off-diagonal elements. The high correlation of the data causes some issues to the data fitting. None the less, we find that either it can be solved by suppressing the largest eigenvalues of the covariance or it is not directly related to the BAO. We test our BAO fitting pipeline using a set of mock catalogs. The data set is dedicated for Dark Energy Survey Year 3 (DES Y3) BAO analyses and includes realistic photo-z distributions. The theory template is in good agreement with mock measurement. Based on the DES Y3 mocks, ξp statistic is forecast to constrain the BAO shift parameter α to be 1.001 ± 0.023, which is well consistent with the corresponding constraint derived from the angular correlation function measurements. Thus, ξp offers a competitive alternative for the photometric data analyses.

Mueller Matrix Ellipsometric Approach on the Imaging of Sub-Wavelength Nanostructures

Conventional spectroscopic ellipsometry is a powerful tool in optical metrology. However, when it comes to the characterization of non-periodic nanostructures or structured fields that are much smaller than the illumination spot size, it is not well suited as it integrates the results over the whole illuminated area. Instead, imaging ellipsometry can be applied. Especially imaging Mueller matrix ellipsometry is highly useful in nanostructure characterization and defect inspection, as it is capable to measure the complete Mueller matrix for each pixel in a microscope image of the sample. It has been shown that these so-called Mueller matrix images can help to distinguish geometrical features of nanostructures in the sub-wavelength regime due to visible differences in off-diagonal matrix elements. To further investigate the sensitivity of imaging Mueller matrix ellipsometry for sub-wavelength sized features, we designed and fabricated a sample containing geometrical nanostructures with lateral dimensions ranging from 50 to 5,000 nm. The structures consist of square and circular shapes with varying sizes and corner rounding. For the characterization of their Mueller matrix images, we constructed an in-house Mueller matrix microscope capable of measuring the full Mueller matrix for each pixel of a CCD camera, using an imaging system and a dual-rotating compensator configuration for the ellipsometric system. The samples are illuminated at 455 nm wavelength and the measurements can be performed in both transmission and reflection. Using this setup, we systematically examine the sensitivity of Mueller matrix images to small features of the designed nanostructures. Within this contribution, the results are compared with traceable atomic force microscopy measurements and the suitability of this measurement technique in optical nanometrology is discussed. AFM measurements confirm that the fabricated samples closely match their design and are suitable for nanometrological test measurements. Mueller matrix images of the structures show close resemblance to numerical simulations and significant influence of sub-wavelength features to off-diagonal matrix elements.

Shortcuts to adiabaticity with general two-level non-Hermitian systems

Shortcuts to adiabaticity are alternative fast processes which reproduce the same final state as the adiabatic process in a finite or even shorter time, which have been extended from Hermitian systems to non-Hermitian systems in recent years, but they are barely explored for general non-Hermitian systems where off-diagonal elements of the Hamiltonian are not Hermitian. In this paper, we propose a shortcuts to adiabaticity technique which is based on a transitionless quantum driving algorithm to realize population transfer for general two-level non-Hermitian systems and give both exact and approximate analytical solutions of the corresponding counteradiabatic driving Hamiltonian, where the latter can be extended to the zeroth-order and first-order terms by applying perturbative theory. We find that the first-order correction term is different from the previous results, which is caused by the non-Hermiticity of the off-diagonal elements. We work out an exact expression for the control function and present examples consisting of a general two-level system with gain and loss to show the theory. The results suggest that the high-fidelity population transfer can be implemented in general non-Hermitian systems by our method, which works even with strong non-Hermiticity and without rotating wave approximation. Furthermore, we show that the general Hamiltonian the off-diagonal elements of which are not conjugate to each other can be implemented in many physical systems with the present experimental technology, such as an atom-light interaction system and whispering-gallery microcavity, which might have potential applications in quantum information processing.