Length Gauge Research Articles

Virial Relations for Electrons Coupled to Quantum Field Modes.

In this work, we present a set of virial relations for many electron systems coupled to both classical and quantum fields, described by the Pauli–Fierz Hamiltonian in dipole approximation and using length gauge. Currently, there is growing interest in solutions of this Hamiltonian because of its relevance for describing molecular systems strongly coupled to photonic modes in cavities and in the possible modification of chemical properties of such systems compared to the ones in free space. The relevance of such virial relations is demonstrated by showing a connection to mass renormalization and by providing an exact way to obtain total energies from potentials in the framework of quantum electrodynamical density functional theory.

An improved method for the investigation of high-order harmonic generation from graphene**Project supported by the National Natural Science Foundation of China (Grant Nos. 11764038, 11864037, 11765018, and 11664035) and the Science Foundation of Northwest Normal University, China (Grant No. NWNU-LKQN-17-1).

High-order harmonic generation (HHG) of bulk crystals in strong laser field is typically investigated with semiconductor Bloch equations (SBEs). However, in the length gauge, it suffers from the divergence for the crystals with a zero band gap, such as graphene, using both Bloch- and Houston-states expansion methods. Here, we present a method of solving the SBEs based on time-dependent Bloch basis, which is equivalent to semiconductor Bloch equations in the velocity gauge. Using this method, we investigate the HHG of a single-layer graphene. It is found that our results for population are in good agreement with the other results. For a initial condition py = 0, we find the electrons just move in single valence band or conduction band, which are in accord with classical results. Our simulations on the HHG dependence of polarization of driving laser pulse confirm that 5th, 7th, and 9th harmonic yields increase to the maximal value when laser ellipticity ε ≈ 0.3. What is more, similar to the case of atoms in the laser field, the total strength of 3rd harmonic decrease monotonically with the increase of ε. In addition, we simulate the dependence of HHG on crystallographic orientation with respect to the polarization direction of linear mid-infrared laser pulse, and the results reveal that for higher harmonics, their radiation along with the change of rotation angle θ reflects exactly the sixfold symmetry of graphene. Our method can be further used to investigate the behaviors of other materials having Dirac points (i.e., surface states of topological insulators) in the strong laser fields.

Calculations of Bethe logarithm for hydrogen and helium using B-splines in different gauges

The efficient and simple B-splines variational method is successfully used to calculate Bethe logarithm for the hydrogen atom in the velocity and length gauges. The ground state Bethe logarithm of hydrogen with fourteen accurate figures is obtained in the velocity gauge, and in the length gauge the ground state value has eleven accurate figures. Present velocity- and length-gauge results for the ns, np, nd, and nf states up to n = 200 of hydrogen are above the 10−10 level of relative difference, which represent the successful variational attempt to calculate Bethe logarithm of hydrogen in the velocity and length gauges. In addition, the B-splines variational method is successfully extended to calculate Bethe logarithm for the helium atom combined with configuration interaction. Results of numerical calculations for n1,3S up to n = 8 states are presented in the acceleration gauge, the velocity gauge, and a hybrid of the velocity and acceleration gauges(also called pa-gauge). For the 23S state, the acceleration-gauge value of 4.364 036 7(2) a.u. and the pa-gauge value of 4.364 036 4(2) a.u. both have eight significant digits. For other triplet S states, present results in three different gauges all have five to seven accurate figures. While for the singlet S states, the best convergent values are obtained in the pa-gauge, of which the relative differences with the latest results are at the 10−6 to 10−8 orders of magnitude.

Structure gauges and laser gauges for the semiconductor Bloch equations in high-order harmonic generation in solids

The semiconductor Bloch equations (SBEs) are routinely used for simulations of strong-field laser-matter interactions in condensed matter. In systems without inversion or time-reversal symmetries, the Berry connections and transition dipole phases (TDPs) must be included in the SBEs, which in turn requires the construction of a smooth and periodic structure gauge for the Bloch states. Here, we illustrate a general approach for such a structure-gauge construction for topologically trivial systems. Furthermore, we investigate the SBEs in the length and velocity gauges, and discuss their respective advantages and shortcomings for the high-harmonic generation (HHG) process. We find that in cases where we require dephasing or separation of the currents into interband and intraband contributions, the length gauge SBEs are computationally more efficient. In calculations without dephasing and where only the total current is needed, the velocity gauge SBEs are structure-gauge independent and are computationally more efficient. We employ two systems as numerical examples to highlight our findings: an 1D model of ZnO and the 2D monolayer hexagonal boron nitride (h-BN). The omittance of Berry connections or TDPs in the SBEs for h-BN results in nonphysical HHG spectra. The structure- and laser-gauge considerations in the current work are not restricted to the HHG process, and are applicable to all strong-field matter simulations with SBEs.

On-machine profile measurement of a micro cutting edge by using a contact-type compact probe unit

A new contact-type on-machine measurement system is designed and developed for the evaluation of a micro cutting edge profile. The measurement system is composed of a compact probe unit having a sharp stylus mounted on a flexible beam, an inner displacement sensor for the detection of the stylus displacement, and a two-axis precision positioning system. For the evaluation of tool faces having a steep slope, a new probing procedure with the enhancement of the inner displacement sensor integrated into the probe unit is newly proposed. After the design and development of the probe unit, the feasibilities of the developed measurement system and the proposed probing procedure are demonstrated through some basic experiments. Regarding the out-of-straightness and angular error motion of the two-axis positioning system employed in the developed measurement system, a pair of length gauges is newly employed to reduce the influences of error motions of the stage system. The topographic profile of the micro cutting edge obtained by the measurement system with the modified probe unit is then compared with those obtained by a commercial stylus profiler and a laser confocal microscope. The feasibility and effectiveness of the developed on-machine tool edge profile measurement system are also demonstrated through uncertainty analysis based on the GUM with the Monte-Carlo method.

Filling gap of combination of gauge and analytical method in KFR-like theory**Project supported by the National Natural Science Foundation of China (Grant Nos. 11274149 and 11304185) and the Program of Shenyang Key Laboratory of Optoelectronic Materials and Technology, China (Grant No. F12-254-1-00).

Bauer recently presented a formula for the ionization rate of a hydrogen atom in a strong linearly polarized laser field [J. Phys. B 49 145601 (2016)]. He started from the Keldysh probability amplitude in the length gauge and utilized Reiss’s method in the velocity gauge. Instead, according to the Reiss probability amplitude in the velocity gauge and Keldysh’s derivation for the length gauge, we derive a formula for the ionization rate of a ground-state hydrogen atom or a hydrogen-like atom in a strong linearly polarized laser field. We compare the numerical results of the total ionization rate and the photoelectron energy distribution calculated from our formula with the results from Keldysh, Reiss, and Bauer. We find that the apparent discrepancies in the ionization rate are caused not only by different gauges, but also by different analytical methods used to derive the ionization rate.

Spectral shift in high-order harmonic generation from periodic potentials

We theoretically investigate the high-order harmonic generation (HHG) of the solid by solving the one-dimensional time-dependent Schrödinger equation in the dipole approximation and length gauge. The spectral shift of the HHG with the solid can be observed. It is illustrated that the contribution to the HHG from the rising and falling parts of the laser pulse is asymmetric, which results in the occurrence of spectral shift. We have also investigated the HHG in laser field with the increase of the laser peak strength; the result shows that the intensity of the harmonic is not continuous, which is illustrated by the electron population.

Comparative analytical and experimental study of fabricated identical surface and interior permanent magnet BLDC motor prototypes

This paper presents a novel and exhaustive investigation involving in-depth analysis, performance evaluation and comparative study of two 0.75 hp, 4-pole, 1500 rpm laboratory prototypes of Brushless DC (BLDC) motors of identical nominal ratings with surface and interior permanent magnet rotor structures having the same stator and winding (integral slot distributed winding). Both the motors were designed and developed in the lab. The major electrical variables (such as rated power, speed, voltage, current, number of poles, etc.) and the stator (such as core material, stator lamination, stack length, winding pattern and wire gauge) of the fabricated prototypes have also been kept identical to pin-point the direct influence of the two different rotor configurations (viz., surface vs interior permanent magnet) on the parameters, performance and operation of these BLDC motors. Additionally, to ensure unbiased basis for appropriate comparison, the overall volumes of magnets/pole in both the motors have also been kept similar. A detailed comparison of different quantities like air-gap flux density distribution, THD in induced voltage, torque ripple, losses and efficiency, torque–speed characteristics with field-weakening capability, steady state parameters at different operating conditions, etc. has been conducted for the said motors and the salient points duly highlighted. The vulnerability of the permanent magnets to demagnetisation based on armature reaction, particularly during a sudden fault, has also been investigated in both the cases. The theoretically determined parameters and analytically evaluated performance figures have been verified through standard FEM packages, and later validated experimentally on the fabricated prototypes. Very good mutual agreement has been observed between predicted and experimental values.

Difference frequency generation in topological semimetals

When two lasers are applied to a non-centrosymmetric material, light can be generated at the difference of the incoming frequencies $\Delta\omega$, a phenomenon known as difference frequency generation (DFG), well characterized in semiconductors. In this work, we derive a general expression for DFG in metals, which we use to show that the DFG in chiral topological semimetals under circular polarized light is quantized in units of $e^3/h^2$ and independent of material parameters, including the scattering time $\tau$, when $\Delta\omega \gg \tau^{-1}$. In this regime, DFG provides a simpler alternative to measure a quantized response in metals compared to previous proposals based on single frequency experiments. Our general derivation unmasks, in addition, a free-carrier contribution to the circular DFG beyond the semiclassical one. This contribution can be written as a Fermi surface integral, features strong frequency dependence, and oscillates with a $\pi/2$ shift with respect to the quantized contribution. We make predictions for the circular DFG of chiral and non-chiral materials using generic effective models, and ab-initio calculations for TaAs and RhSi. Our work provides a complete picture of the DFG in the length gauge approach, in the clean, non-interacting limit, and highlights a plausible experiment to measure topologically quantizated photocurrents in metals.

IDENTIFICATION OF CONDITION OF LINEAR DISPLACEMENT SENSOR

Urgency of the research. There are several types of displacement sensor available on market. Displacement sensor investigated in this work is based on optical encoder principle. Condition of sensors changes with using of it. Periodically it is necessary to check its condition, if it is within the declared limits. Target setting. Displacement linear sensor is mounted in comparator stand for the verifying of its condition using the set of length gauge blocks. Length gauge blocks allows to set up the etalon of length with various dimension in interval from 0.5mm to 100 mm. The systematic errors of used set of length gauge blocks of grade “0” are very small in comparing with measured dimension and measured deviations. Actual scientific researches and issues analysis. It is necessary to check actual status of sensor. It means that verification process will obtain the information about maximum permissible error and information about reliability. Uninvestigated parts of general matters defining. The main problem was to identify condition of sensor. The question of the probability distribution of measured values and uncertainty balance are uninvestigated, because the next research will be focused to this are. The research objective. The aim is to obtain maximum permissible error of explored sensor. On the base of deviation of measurement made on length gauges could be expressed. The optimal number of measurement is problem to know, because the low number will cause big uncertainty of measurement and large number of measurement will cause the large cost of measurement. The statement of basic materials. Gauge length blocks have been used for verification of investigated sensor. The set of gauge length blocks of grade “0” has been used, which are preferred mainly for calibration or verification purposes. Maximum permissible error has been estimated as math model for next using. Also optimal number of measurement is identified from analysis of standard deviation of measurements made one hundred times on selected dimensions. Conclusions. The investigated sensor meets the maximum permissible error limits set by the manufacturer with a large margin, and so the maximum permissible error limits have been tightened so that the measurement uncertainty is better. The sensor can be used in dimensional measurement applications, even in industrial conditions.

Numerical solution of three-dimensional time-dependent Schrödinger equation based on graphic processing unit acceleration

In the field of quantum mechanics, the theoretical study of the interaction between intense laser field and atoms and molecules depends very much on the numerical solution of the time-dependent Schrödinger equation. However, solving the three-dimensional time-dependent Schrödinger equation is not a simple task, and the analytical solution cannot be obtained, so it can only be solved numerically with the help of computer. In order to shorten the computing time and obtain the results quickly, it is necessary to use parallel methods to speed up computing. In this paper, under the background of strong field ionization, the three-dimensional time-dependent Schrödinger equation of hydrogen atom is solved in parallel, and the suprathreshold ionization of hydrogen atom under the action of linearly polarized infrared laser electric field is taken for example. Based on the spherical polar coordinate system, the time-dependent Schrödinger equation is discretized by the splitting operator-Fourier transform method, and the photoelectron continuous state wave function under the length gauge can be obtained. In Graphics processing unit (GPU) accelerated applications, the sequential portion of the workload runs on central processing unit (CPU) (which is optimized for single-threaded performance), while the compute-intensive part of the application runs in parallel on thousands of GPU cores. The GPU can make full use of the advantage of fine-grained parallelism based on multi-thread structure to realize parallel acceleration of the whole algorithm. Two accelerated computing modes of CPU parallel and GPU parallel are adopted, and their parallel acceleration performance is discussed. Compared with the results from the existing physical laws, the calculation error is also within an acceptable range, and the result is also consistent with the result from the existing physical laws of suprathreshold ionization, which also verifies the correctness of the program. In order to obtain a relatively accurate acceleration ratio, many different experiments are carried out. Computational experiments show that under the condition of ensuring accuracy, the GPU parallel computing speeds by up to about 60 times maximally based on the computational performance of CPU. It can be seen that the accelerated numerical solution of three-dimensional time-dependent Schrödinger equation based on GPU can significantly shorten the computational time. This work has important guiding significance for rapidly solving the three-dimensional time-dependent Schrödinger equation by using GPU.

Gauge-invariant time-dependent configuration interaction singles method: applications to high-harmonic generation from atoms

SynopsisWe present gauge-invariant reformulation of time-dependent configuration interaction singles (TDCIS) method and its applications to high-harmonic generation (HHG) from three-dimensional atoms. The conventional TDCIS method suffers from a violation of gauge-invariance, which prevents efficient simulations of high-field phenomena within the velocity gauge. In our reformulation, rotated velocity gauge transformed from the length gauge is employed instead of the velocity gauge. We simulate HHG from atoms and show gauge-invariance and performance of our rotated velocity gauge.

Theoretical calculations of nonlinear optical calculations of 2D materials

One important feature of two dimensional (2D) materials is that they possess an exceptional nonlinear optical (NLO) response to light, with conduc¬tivities that are several orders of magnitude larger than their 3D counterparts. The theoretical descriptions of these NLO responses in crystalline systems in¬volve two different representations of the perturbation: the length and velocity gauges. The former has been the formalism of choice for the past two decades; the latter was implemented only recently, due to concerns that it could not be pratically implemented without breaking sum rules – a set of identities that en¬sure the equivalence between the two formalisms – which would then render the results unphysical. In this work, we shall review and summarize our contri¬butions to the study of the two formalisms and of their relationship by means of the aforementioned sum rules.

The software instrument for gymnast posture analysis

The general aim of this research was to develop an ideal instrument to analyse the ideal anthropometric posture of artistic gymnastic sports so that profiles and proportions of the length of the athletes were ideal through analysis images. Research and development methods used the following paths for problem identification, information gathering, product design, design validation, product testing, trial used and mass product manufacturing. The results of the study contain software design (UNG) with the performance of storing and analysing images by adjusting the real coordinate scale and height digital scale as a reference to measure the athlete's body length and the degree of posture imbalance. Validity was measured in two ways, namely correlating the standard length gauge No. Reg C4 / VI / 250419 with measurements in the software amounting to 0.92 very high categories and averaging in hundreds of assessments of 92 ICT experts and sports experts at 96.5. Reliability using retest was .95 very high category. Posture Image Analysis of 150 male world gymnasts All the figures around depict the average height, Percentage of limb length, posture imbalance. in degree, posture analysis which is suitable and efficient in displaying for each exercise.it describe an average height of 166 cm with a leg length proportion of 54%, Body 26%, Head 13% and neck 5%. The conclusion of UNG software was to effectively analyse the profile image of the sample male artistic gymnastic ideal to produce a profile and the length of anthropometric proportions of the athlete's body members in accordance with the actual object.

Comparative study on atomic ionization in bicircular laser fields by length and velocity gauges S-matrix theory**Project supported by the Key Laboratory Project of Computational Physics of National Defense Science and Technology of China (Grant No. 6142A05180401), the National Key Program for S&T Research and Development of China (Grant Nos. 2019YFA0307700 and 2016YFA0401100), and the National Natural Science Foundation of China

Ionization of atoms in counter-rotating and co-rotating bicircular laser fields is studied using the S-matrix theory in both length and velocity gauges. We show that for both the bicircular fields, ionization rates are enhanced when the two circularly polarized lights have comparable intensities. In addition, the curves of ionization rate versus the field amplitude ratio of the two colors for counter-rotating and co-rotating fields coincide with each other in the length gauge case at the total laser intensity 5 × 1014 W/cm2, which agrees with the experimental observation. Moreover, the degree of the coincidence between the ionization rate curves of the two bicircular fields decreases with the increasing field amplitude ratio and decreasing total laser intensity. With the help of the ADK theory, the above characteristics of the ionization rate curves can be well interpreted, which is related to the transition from the tunneling to multiphoton ionization mechanism.

Many-body calculations of two-photon, two-color matrix elements for attosecond delays

We present calculations for attosecond atomic delays in photoionization of noble gas atoms based on full two-color two-photon Random-Phase Approximation with Exchange in both length and velocity gauge. Gauge invariant atomic delays are demonstrated for the complete set of diagrams. The results are used to investigate the validity of the common assumption that the measured atomic delays can be interpreted as a one-photon Wigner delay and a universal continuum--continuum contribution that depends only on the kinetic energy of the photoelectron, the laser frequency and the charge of the remaining ion, but not on the specific atom or the orbital from which the electron is ionized. Here we find that although effects beyond the universal IR--photoelectron continuum--continuum transitions are rare, they do occur in special cases such as around the $3s$ Cooper minimum in argon. We conclude also that in general the convergence in terms of many-body diagrams is considerably faster in length gauge than in velocity gauge.

Implementation of a gauge-invariant time-dependent configuration-interaction-singles method for three-dimensional atoms

We present a numerical implementation of the gauge-invariant time-dependent configuration interaction singles (TDCIS) method [Appl. Sci. 8, 433 (2018)] for three-dimensional atoms. In our implementation, orbital-like quantity called channel orbital [Phys. Rev. A 74, 043420 (2006)] is propagated instead of configuration-interaction (CI) coefficients, which removes a computational bottleneck of explicitly calculating and storing numerous virtual orbitals. Furthermore, besides its physical consistency, the gauge-invariant formulation allows to take advantages of the velocity gauge treatment of the laser-electron interaction over the length gauge one in the simulation of high-field phenomena. We apply the present implementation to high-harmonic generation from helium and neon atoms, and numerically confirms the gauge invariance and demonstrates the effectiveness of the rotated velocity gauge treatment.

Data Acquisition System Design of Length Gauge

Length gauge is one of measurement standard in the Length Metrology Laboratory, Research Center Metrology – LIPI. Practical use and good accuracy make it a mainstay in calibration. In use, length gauge requires a display unit to display readings of measuring results. The display unit is very susceptible to damage because it is directly connected to an electric power supply. To procure the display unit, requires funds and time. Therefore, the objective of the research is to design a length gauge data acquisition system that is simple and low cost without compromising function. Research was carried out by making a current into a voltage converter, converting an analog voltage signal into a pulse signal, separates the signal when the length gauge moves up and down, counting the generated pulse signal and displays it on the PC. Once the data acquisition system is calibrated, it was found that the readability of the system is 2 μm and with an uncertainty of 2.0 m. With this value of this uncertainty, based on ANSI/NCSL Z540-1-1994, the data acquisition system can be used as standard to calibrate measuring devices with an accuracy of at least 8 m.

Pyridinium Crosslinks (Pyd) in the Urine is Associated with Stunting in Neonates

Aims: The aim of the research is to evaluate the reliability of bone resorption biomarkers called Pyridinium Crosslinks (Pyd) in the urine of the neonates as an evaluation to bone growth of the neonate, as an indicator of stunting.
 Study Design: A cross-sectional study.
 Place and Duration of Study: Andini Mothers and Children Hospital (Pekanbaru, Indonesia). Duration of the study was between, August until September 2014.
 Methodology: Subjects of study were 35 healthy neonates. Subjects were recruited at the first 3 days of life. Body length gauges, digital weighting scale, family socioeconomic questionnaires and Pyd kit were used to collect the data. Differences in the mean of the research variables were tested using an Independent t-test. 
 Results: Results showed that there were significant differences (p<0.01) in terms of height for age and Pyd in the urine of stunted (body length <48 cm) versus normal (body length ≥48 cm) neonates. The contents of Pyd in the urine of stunted neonates were 982.9±61.6 and normal neonates was 594.1±266.1 nmol/mmol.
 Conclusion: Therefore, there is a possible association between height for age and Pyd in the urine as a potential early indicators to identify stunted and normal neonates.

Excited States of Molecules in Strong Uniform and Nonuniform Magnetic Fields.

This paper reports an implementation of Hartree-Fock linear response with complex orbitals for computing electronic spectra of molecules in strong external magnetic fields. The implementation is completely general, allowing for spin-restricted, spin-unrestricted, and general two-component reference states. The method is applied to small molecules placed in strong uniform and nonuniform magnetic fields of astrochemical importance at the Random Phase Approximation level of theory. For uniform fields, where comparison is possible, the spectra are found to be qualitatively similar to those recently obtained with equation of motion coupled cluster theory. We also study the behavior of spin-forbidden excitations with progressive loss of spin symmetry induced by nonuniform magnetic fields. Finally, the equivalence of length and velocity gauges for oscillator strengths when using complex orbitals is investigated and found to hold numerically.