Classical Transition Research Articles

Synthesis and redox catalysis of Carbodiphosphorane ligated stannylene.

Heavier group 14 carbene analogues, exhibiting transition-metal-like behavior, display remarkable capability for small molecule activation and coordination chemistry. However, their application in redox catalysis remains elusive. In this paper, we report the synthesis and isolation of a stannylene with carbodiphosphorane ligand. The nucleophilic reactivity at the divalent tin center is elucidated by computational and reactivity studies. Moreover, this stannylene exhibits catalytic activity in the hydrodefluorination reaction of fluoroarenes. Mechanistic investigations into the elementary steps confirm a SnII/SnIV redox cycle involving C-F oxidative addition, F/H ligand metathesis, and C-H reductive elimination. This low-valent SnII catalytic system resembles the classical transition metal catalysis. Notably, this represents metallomimetic redox catalysis utilizing carbene analogue with heavier group 14 element as a catalyst.

Manipulating the EphB4-ephrinB2 axis to reduce metastasis in HNSCC.

The EphB4-ephrinB2 signaling axis has been heavily implicated in metastasis across numerous cancer types. Our emerging understanding of the dichotomous roles that EphB4 and ephrinB2 play in head and neck squamous cell carcinoma (HNSCC) poses a significant challenge to rational drug design. We find that EphB4 knockdown in cancer cells enhances metastasis in preclinical HNSCC models by augmenting immunosuppressive cells like T regulatory cells (Tregs) within the tumor microenvironment. EphB4 inhibition in cancer cells also amplifies their ability to metastasize through increased expression of genes associated with hallmark pathways of metastasis along with classical and non-classical epithelial-mesenchymal transition. In contrast, vascular ephrinB2 knockout coupled with radiation therapy (RT) enhances anti-tumor immunity, reduces Treg accumulation into the tumor, and decreases metastasis. Notably, targeting the EphB4-ephrinB2 signaling axis with the engineered ligands ephrinB2-Fc-His and Fc-TNYL-RAW-GS reduces local tumor growth and distant metastasis in a preclinical model of HNSCC. Our data suggests that targeted inhibition of vascular ephrinB2 while avoiding inhibition of EphB4 in cancer cells could be a promising strategy to mitigate HNSCC metastasis.

Influence of large-scale free-stream turbulence on bypass transition in air and organic vapour flows

The free-stream turbulence (FST) induced transition in perfect and non-ideal gas zero-pressure-gradient flat-plate boundary layers is investigated by means of large-eddy simulations. The study focuses on the influence of large incoming disturbances over the laminar-to-turbulent transition, by comparing two different integral length scales $L_f$ , which differ by a factor of seven, at different FST intensities $T_u$ . High-subsonic dense-gas boundary layers of the organic vapour Novec649, representative of organic Rankine cycle applications, are compared with air flows at Mach numbers 0.1 and 0.9. Compressibility and non-ideal gas effects are shown to be of minor importance in comparison to the influence of the FST integral length scale $L_f$ . An increase of the inlet turbulent intensity always promotes transition, whereas an increase of $L_f$ has a double effect on the transition onset. At $T_u=2.5\,\%$ , increasing $L_f$ promotes the transition, while it tends to delay transition for an FST intensity of 4 %. Larger FST integral scales tend to increase the spanwise distance between laminar streaks generated in the boundary layer. Two competing transition scenarios are observed. When the incoming turbulence intensity and length scale are moderate, the classical bypass route consists in the linear non-modal growth of streaks, which then experience secondary instabilities (sinuous or varicose) and lead to the generation of turbulent spots. The second scenario is characterized by the appearance of $\Lambda$ -shaped structures near the inlet, which are further stretched to hairpin vortices before breaking down to turbulence. Spot inceptions can therefore occur at earlier locations than the streak growth. We are then faced with a competition between the classical bypass transition and nonlinear response mechanisms that ‘bypass’ this route. The present case at high $L_f$ and low $T_u$ is an example of a competing scenario, but even for the higher $T_u$ and $L_f$ conditions, only approximately one-third of turbulent spots are due to the $\Lambda$ -shaped events. The nonlinear alternative route has strong similarities with scenarios described previously in the literature in the presence of leading edge effects or due to passing wakes. Such a path is governed by the turbulence intensity, but also by the integral length scale, with both parameters playing a critical role in the generation of the $\Lambda$ -shaped structures near the inlet. This alternative mechanism is found to be robust under varying flow and thermodynamic conditions.

Theoretical Kinetics of Radical-Radical Reaction NH2ṄH + ṄH2 and Its Implications for Monomethylhydrazine Pyrolysis Mechanism.

Significant discrepancies were observed between the experiments and the simulations for ṄH2 time-histories in monomethylhydrazine pyrolysis with the robust mechanism proposed by Pascal and Catoire. The rate of formation analyses for ṄH2 indicated the significance of the reaction NH2ṄH + ṄH2 = H2NN + NH3, which has not been well-defined. In this study, ab initio calculations were performed for the theoretical description of the NH2ṄH + ṄH2 chemistry. Most stationary points on the potential energy surface were quantified at the CCSD(T)/CBS//M06-2X/aug-cc-pVTZ level, and the multireference methods were employed for barrier-less reaction and some transition states. The temperature- and pressure-dependent rate coefficients were determined using classical and microcanonical variational transition state theories. Four primary reaction channels were identified as competitive: 1) The H atom abstraction reaction yielding N2H2(T) + NH3, dominating at 1350-3000 K across the 0.001-100 atm pressure range. 2) The H atom abstraction reaction forming N2H2(S) + NH3, dominating at 800-1350 K and competing with the processes of chemical activation and collisional stabilization below 800 K. 3) The chemical-activated reaction resulting in H2NN(S) + NH3, dominating below 800 K at 0.001 atm. 4) The collisional-stabilized recombination reaction leading to N3H5, becoming significant as pressure increases and dominating below 600 and 650 K at 1 and 100 atm, respectively. The implications of newly calculated NH2ṄH + ṄH2 kinetics for the monomethylhydrazine pyrolysis mechanism were evaluated, and updates were implemented. Sensitivity analyses indicated the necessity of additional research efforts to comprehend the dynamics of CH3NH2 unimolecular and N2H2 + ṄH2 reaction systems. The rate coefficients presented in this study can be employed to develop the chemical kinetic model of nitryl-containing systems.

Hydrogen Evolution Reaction: Log j0 Versus Work Function Φ As Viewed from Transition State Theory As an Eyring Plot

In 1972, Trasatti correlated the log of the exchange current density (j0) for the hydrogen evolution reaction (HER) with the work functions of 31 metal electrodes to yield two parallel regression lines. [1] The lines are separated by Δ log j0 = 3. Thermodynamically, work function 𝚽 is the energy to remove an electron from a metal to infinity. Perhaps counterintuitively, platinum group metals (PGMs) with the highest 𝚽 sustain the highest j0 values.The elementary, interfacial electron transfer occurs between the electrode and adsorbed hydrogen cation and adsorbed hydrogen atom.H+ ads + e ⇌ H● ads A materials dependent rate expression is developed as follows. [2]The properties of the metal are introduced explicitly into reaction mechanism, characterized by M(e) and M(0), the metal electrode with and without the electron.M(e) + H+ ads ⇄ M(0) + H● ads Classical transition state theory (TST) characterizes the free energy of activation ΔG‡ as the energy difference between the transition state (TS) and the minimum energy of the reactants. The TS forms at equilibrium, where adsorbed hydrogen and metal electrode share the electron.M(e) + H+ ads ⇌ [M ●●● e ●●● Hads]‡ ⇌ M(0) + H● ads Electrochemical potentials set the energy difference in the products and reactant, which sets ΔG‡. Chemical potential (𝞵) difference for the metal without and with the electron define 𝚽. F𝚽 = 𝞵M(0) - 𝞵M(e) F𝚽 is further defined as a chemical free energy that that lowers ΔG‡. F𝚽 is a material specific property that is atypically specified in the rate equation. Because log j0 ∝ - ΔG‡, and chemical free energy F𝚽 lowers ΔG‡, log j0 ∝ F𝚽. Thus, j0 increases exponentially with F𝚽, so PGMs with highest 𝚽 sustain highest j0. Further, the rate expression embeds material specific property 𝚽.The classical Eyring perspective for generic rate constant k’ is k’ = A’ exp[-ΔG‡/RT]. Rate constant is typically determined by varying temperature in a plot of ln k’ versus T-1, where ln k’ = ln A’ - ΔG‡/RT. For isothermal measurements, log j0 is linear with 𝚽 because F𝚽 is a term ΔG‡ chem that is a component in the total ΔG‡. For metals, where 𝚽 > 0, F𝚽 always lowers ΔG‡ for HER; the total activation energy is lowered in proportion to F𝚽. The slopes of log j0 versus 𝚽 is proportional to 𝝰𝚽F𝚽/2.303, where partition coefficient 𝝰𝚽 characterizes ∂ΔG‡/∂𝚽. For the parallel lines in Trasatti’s plot, 𝝰𝚽 = 0.38. The intercepts are independent of 𝚽.The material specific rate expressions derived from classical TST and electrochemical potentials may provide guidance on design of electrocatalysts and provide a path to integration of classical and contemporary modeling of important reaction processes such as HER.

Quantum Physics, Digital Computers, and Life from a Holistic Perspective

Quantum physics is a linear theory, so it is somewhat puzzling that it can underlie very complex systems such as digital computers and life. This paper investigates how this is possible. Physically, such complex systems are necessarily modular hierarchical structures, with a number of key features. Firstly, they cannot be described by a single wave function: only local wave functions can exist, rather than a single wave function for a living cell, a cat, or a brain. Secondly, the quantum to classical transition is characterised by contextual wave-function collapse shaped by macroscopic elements that can be described classically. Thirdly, downward causation occurs in the physical hierarchy in two key ways: by the downward influence of time dependent constraints, and by creation, modification, or deletion of lower level elements. Fourthly, there are also logical modular hierarchical structures supported by the physical ones, such as algorithms and computer programs, They are able to support arbitrary logical operations, which can influence physical outcomes as in computer aided design and 3-d printing. Finally, complex systems are necessarily open systems, with heat baths playing a key role in their dynamics and providing local arrows of time that agree with the cosmological direction of time that is established by the evolution of the universe.

Bayesian inference for near-field interferometric tests of collapse models

We explore the information which proposed matterwave interferometry experiments with large test masses can provide about parameterizable extensions to quantum mechanics, such as have been proposed to explain the apparent quantum to classical transition. Specifically, we consider a matterwave near-field Talbot interferometer and Continuous Spontaneous Localization (CSL). Using Bayesian inference we compute the effect of decoherence mechanisms including pressure and blackbody radiation, find estimates for the number of measurements required, and provide a procedure for optimal choice of experimental control variables. We show that in a simulated space-based experiment it is possible to reach masses of ∼109u, and we quantify the bounds which can be placed on CSL. These specific results can be used to inform experimental design, and the general approach can be applied to other parameterizable models. Published by the American Physical Society 2024

Expanding wave packets and the quantum to classical transition

The standard description of the transition from quantum to classical mechanics presented in most text books is the proof that the quantum expectation values of position and momentum obey equations of Newtonian form. This is the Ehrenfest theorem. It is combined with the requirement that wave packets remain localised to describe a single particle moving according to classical mechanics. Hence, the natural spreading of wave packets is viewed as a quantum effect. In contradiction to this view, here it is argued that the spreading, where different momentum components separate, is the signature of the quantum to classical transition. The asymptotic spatial wave function becomes proportional to the initial momentum space wave function, which mirrors exactly the well-known far-field diffraction pattern in optics. Trajectories, defined as the locus of the normals to the expanding wave front, are used to illustrate the transition from quantum to classical motion. Again this is the analogue of the wave to beam transition in optics. It is suggested that this analysis of the quantum to classical transition should be incorporated routinely into introductory quantum mechanics courses.

Reentrant phase transitions of quantum black holes

We show that backreaction of quantum fields on black hole geometries can trigger new thermal phase transitions. Specifically, we study the phase behavior of the three-dimensional quantum-corrected static Bañados-Teitelboim-Zanelli black hole, an exact solution to specific semiclassical gravitational equations due to quantum conformal matter, discovered through braneworld holography. Focusing on the canonical ensemble, for large backreaction, we find novel reentrant phase transitions as the temperature monotonically increases, namely, from thermal anti–de Sitter space to the black hole and back to thermal anti–de Sitter. The former phase transition is first-order, a quantum analog of the classical Hawking-Page phase transition, while the latter is zeroth order and has no classical counterpart. Published by the American Physical Society 2024

Tungsten Oxide Dispersed on Silica as Robust and Readily Available Oxo/Imido Heterometathesis Catalyst.

Continuing our investigation of catalytic oxo/imido heterometathesis as novel water-free method for C=N bond construction, we report here the application of classical transition metal oxides dispersed on silica (MOx/SiO2, M=V, Mo, W) as cheap, robust and readily available alternative to the catalysts prepared via Surface Organometallic Chemistry (SOMC). The oxide materials demonstrated activity in heterometathetical imidation of ketones, WO3/SiO2 being the most efficient. We also describe a new well-defined supported W imido complex (≡SiO)W(=NMes)2(Me2Pyr) (Mes=2,4,6-Me3C6H2, Me2Pyr=2,5-dimethylpyrrolyl) and characterize it with SOMC protocols, which allowed us to identify the position of W on the oxo/imido heterometathesis activity scale (Mo<W<Ta) and showed that W in fact provides an optimal balance between activity and stability. Noteworthy, the performance of WO3/SiO2 was only slightly inferior to that of the SOMC analog enabling the efficient synthesis of a broad range of ketimines hardly accessible by classical methods, in most cases W being a competitive and robust alternative to the most active albeit highly sensitive Ti imido systems. Combined with the availability of the catalyst, this facilitates a more widespread application of heterometathesis methodology in organic synthesis including potential larger scale applications.

Entanglement in cosmology

We compute the evolution of the entanglement entropy for a massless field within a spherical region throughout the inflationary period and the subsequent era of radiation domination, starting from the Bunch-Davies vacuum. In order to focus on the entanglement of modes that are directly accessible to observations, we impose an ultraviolet cutoff set by the wavelength of the last mode that exited the horizon at the end of inflation. The transition of each mode towards a squeezed state upon horizon exit during inflation and the additional squeezing when radiation domination sets in enhance the entanglement entropy. Shortly after the transition to the radiation-dominated era, a volume term develops and becomes the leading contribution to the entropy at late times, as is common for systems lying in squeezed states. We estimate the magnitude of the entropy and discuss its interpretation in the light of the quantum to classical transition for modes exiting the horizon during inflation. Our results raise the possibility that the quantum nature of weakly interacting fields, such as gravitational waves resulting from tensor modes during inflation, may be detectable in today's universe. On the other hand, an observer with no knowledge of the degrees of freedom beyond the horizon would interpret the entropy as thermal. From this point of view, the reheating after inflation would be a result of quantum entanglement.

Decoherence and entropy generation at one loop in the inflationary de Sitter spacetime for Yukawa interaction

Abstract The decoherence mechanism is believed to be possibly connected to the quantum to classical transition of the primordial cosmological perturbations in the early universe. In this paper, we extend our previous analysis on decoherence in a fermion and scalar quantum field theory coupled via the Yukawa interaction in the Minkowski spacetime, to the inflationary de Sitter background. We treat the scalar field as the system and the fermions as the environment, and both the fields are taken to be massless. We utilise a non-equilibrium effective field theory formalism, suitable for open quantum systems such as this. We assume that an observer measures only the Gaussian 2-point correlator for the scalar field, as the simplest realistic scenario. In order to compute the von Neumann entropy generated at late times as a measure of the decoherence, we construct the one loop renormalised Kadanoff-Baym equation, which is the equation of motion satisfied by the 2-point correlators in the closed time path Schwinger-Keldysh formalism. These equations account to the self energy corrections. Using this, we next construct the one loop corrected statistical propagator for the scalar, which is related to its phase space area, to compute the von Neumann entropy. We also compute the variation of the von Neumann entropy with respect to relevant parameters. We note the qualitative similarity between our findings and the scenario where both the system and the environment are scalars. Our result is also qualitatively similar to an earlier one found by using the influence functional technique for a massive Yukawa theory.

Evolution of structural and electronic properties standardized description in rhenium disulfide at the bulk-monolayer transition

The structural and electronic properties of ReS2 different forms — three-dimensional bulk and two-dimensional monolayer — were studied within density functional theory and pseudopotentials. A method for standardizing the description of bulk unit cells and "artificial" slab unit cells for DFT research has been proposed. The preference of this method for studying zone dispersion has been shown. The influence of the vacuum layer thickness on specified special high-symmetry points is discussed. Electron band dispersion in both classical 3D Brillouin zones and transition to 2D Brillouin zones in the proposed two-dimensional approach using the Niggli form of the unit cell was compared. The proposed two-dimensional approach is preferable for low-symmetry layered crystals such as ReS2. It was established that the bulk ReS2 is a direct gap semiconductor (band gap of 1.20 eV), with the direct transition lying in the X point of the first Brillouin zone, and it is in good agreement with published experimental data. The reduction in material dimension from bulk to monolayer was conducted with an increasing band gap up to 1.45 eV, with a moving direct transition towards the Brillouin zone center. The monolayer of ReS2 is a direct-gap semiconductor in a wide range of temperatures, excluding only a narrow range at low temperatures, where it comes as a quasi-direct gap semiconductor. The transition, situated directly in the Γ-point, lies 3.3 meV below the first direct transition located near this point. The electronic density of states of ReS2 in the bulk and monolayer cases of ReS2 were analyzed. The molecular orbitals were built for both types of ReS2 structures as well as the electron difference density maps. For all types of ReS2 structures, an analysis of populations according to Mulliken and Voronoi was carried out. All calculated data is discussed in the context of weak quantum confinement in the 2D case.

Quantum symmetrization transition in superconducting sulfur hydride from quantum Monte Carlo and path integral molecular dynamics

AbstractWe study the structural phase transition, originally associated with the highest superconducting critical temperature Tc measured in high-pressure sulfur hydride. A quantitative description of its pressure dependence has been elusive for any ab initio theory attempted so far, raising questions on the actual mechanism leading to the maximum of Tc. Here, we estimate the critical pressure of the hydrogen bond symmetrization in the Im$$\bar{3}$$ 3 ¯ m structure, by combining density functional theory and quantum Monte Carlo simulations for electrons with path integral molecular dynamics for quantum nuclei. We find that the Tc maximum corresponds to pressures where local dipole moments dynamically form on the hydrogen sites, as precursors of the ferroelectric Im$$\bar{3}$$ 3 ¯ m-R3m transition, happening at lower pressures. For comparison, we also apply the self-consistent harmonic approximation, whose ferroelectric critical pressure lies in between the ferroelectric transition estimated by path integral molecular dynamics and the local dipole formation. Nuclear quantum effects play a major role in a significant reduction (≈50 GPa) of the classical ferroelectric transition pressure at 200 K and in a large isotope shift (≈25 GPa) upon hydrogen-to-deuterium substitution of the local dipole formation pressure, in agreement with the corresponding change in the Tc maximum location.

Computational Research on Ag(I)-Catalyzed Cubane Rearrangement: Mechanism, Metal and Counteranion Effect, Ligand Engineering, and Post-Transition-State Desymmetrization.

Ag(I) salts have demonstrated superior catalytic activity in the cubane-cuneane rearrangement. This research presents a comprehensive mechanistic investigation using high-level computations. The reaction proceeds via oxidative addition (OA) of Ag(I) to the C-C bond, followed by C-Ag bond cleavage and subsequent dynamically concerted carbocation rearrangement. The OA of Ag(I) exhibits significant more electrophilic nature than classical transition metal-induced OA, and the superior catalytic activity of Ag(I) is attributed to the accessibility of a highly electrophilic "bare" Ag+ center and a relatively weak Ag-C bond. However, the highly Lewis acidic nature of the Ag(I) center limits the substrate scope. To address this problem, ligand and counteranion screening was conducted, revealing that chiral biarylether ligands in combination with BF4- as the counteranion offer both enhanced reactivity and improved chemoselectivity while suppressing the Lewis acidity. Additionally, quasi-classical molecular dynamics simulations indicate the possibility of a novel desymmetrization pathway through post-transition-state dynamics in the biarylether-Ag(I)-BF4- system, thereby providing a potential avenue for enantioselective cuneane synthesis.

Towards the generation of mechanical Kerr-cats: awakening the perturbative quantum Moyal corrections to classical motion

The quantum to classical transition is determined by the interplay of a trio of parameters: dissipation, nonlinearity, and macroscopicity. Why is nonlinearity needed to see quantum effects? And, is not an ordinary pendulum quite nonlinear already? In this manuscript, we discuss the parameter regime where the dynamics of a massive oscillator should be quantum mechanical in the presence of dissipation. We review the outstanding challenge of the dynamical generation of highly quantum mechanical cat states of a massive ‘pendulum’, known as Kerr-cats. We argue that state-of-the-art cold atom experiments may be in a position to reach such a nonlinear regime, which today singles out superconducting quantum circuits. A way to stabilize Schrödinger cat superpositions of a mechanical atomic oscillator via parametric squeezing and further protected by an unusual form of quantum interference is discussed. The encoding of a neutral atom Kerr-cat qubit is proposed.

Synthesis of a measurement procedure for estimating the orientation of a small unmanned aerial vehicle under conditions of changing status of measurement results

The problem of processing measurement information with changing status of measurement results of micromechanical sensors of an intelligent on-board measuring system of a small unmanned aerial vehicle is considered. While vehicle is in the air, the measurement results status changes from confirmed to orienting, for example, due to sensor defects, degradation of the measuring channel, the appearance of false measurement output signals of micromechanical sensors due to vibrations and shocks caused by the movement of air masses. As a result, the probability of stability loss of a small unmanned aerial vehicle increases and it is necessary to raise the accuracy of estimating its orientation in conditions of changing status of measurement results. The measuring procedure of the Kalman structure is considered, the equations of which are determined with accuracy to the parameters of the transition and noise matrices of the state, as well as the perturbation vector, characterizing the measuring process. The parameter values are determined by a mathematical model for converting measuring information based on a dynamic mathematical model, which distinguishes the developed measuring procedure from a measuring procedure with a classical transition matrix. A neural network is used to find unknown parameters. A multilayer perceptron was selected as the basis of a neural network, for which an error back propagation algorithm is used to train. Based on the results of mathematical modeling and measurement experiment, it was found that the accuracy of the synthesized measuring procedure based on a dynamic mathematical model is higher than the accuracy of the measuring procedure of the Kalman structure with a classical transition matrix. The results of the study will be useful in the development of intelligent measurement procedures for on-board measurement systems operating under conditions of changing status of measurement results

Nanoparticle Interferometer by Throw and Catch

Matter wave interferometry with increasingly larger masses could pave the way to understanding the nature of wavefunction collapse, the quantum to classical transition, or even how an object in a spatial superposition interacts with its gravitational field. In order to improve upon the current mass record, it is necessary to move into the nanoparticle regime. In this paper, we provide a design for a nanoparticle Talbot–Lau matter wave interferometer that circumvents the practical challenges of previously proposed designs. We present numerical estimates of the expected fringe patterns that such an interferometer would produce, considering all major sources of decoherence. We discuss the practical challenges involved in building such an experiment, as well as some preliminary experimental results to illustrate the proposed measurement scheme. We show that such a design is suitable for seeing interference fringes with 106 amu SiO2 particles and that this design can be extended to even 108 amu particles by using flight times below the typical Talbot time of the system.

Superquantum effects on physical systems from a hydrodynamic perspective

Employing a nonlinear time-dependent Schrödinger equation that describes the transition between quantum and classical regimes, we explore superquantum effects on physical systems from a hydrodynamic point of view. The degree of quantumness, a pivotal parameter in this transition equation, gauges the impact of quantum effects arising from the quantum potential. As this parameter varies continuously between zero and one, it facilitates the gradual transformation of systems from purely classical to entirely quantum regimes. We extend our analysis by setting the degree of quantumness greater than one, thereby investigating superquantum phenomena. Notably, this transition equation can be reframed as a scaled time-dependent Schrödinger equation with a rescaled Planck constant. Through computational simulations derived from the scaled Schrödinger equation, we examine superquantum implications for two systems, including Gaussian wave packet scattering from an Eckart barrier and metastable state decay. This research suggests that the hydrodynamic formulation of the quantum–classical transition equation offers a profound understanding of superquantum influences on physical systems.

Quantum nonlocality evolution for two entangled mesoscopic fields under decoherence

Investigation of the nonlocality evolution of entangled mesoscopic fields under decoherence not only is important for understanding the quantum–classical transition, but also has relevance to quantum communication and quantum computation protocols based on continuous variables. According to previous formulations of Bell inequalities, the system loses nonlocal features far before the disappearance of entanglement. We here construct a new version of Bell signal based on rotated and displaced on–off correlations, with which the Bell inequality is violated as long as there remains entanglement and the field state components keep quasiorthogonal. Consequently, the nonlocal character revealed by our formulation decays much slower compared with those based on previous ones. More importantly, there exists a wide regime where the Bell inequality is restored with previous formulations but remains to be violated based on our correlation operators.