Lattice Sites Research Articles

Threshold displacement in the tungsten-carbon system

ABSTRACT Threshold displacement energies (Ed) of carbon and tungsten in tungsten carbide (WC), W2C, tungsten and diamond are predicted using molecular dynamics. The spatial dependence of Ed is probed by considering a geodesic projection of a symmetrically distinct arc of crystallographic directions for each lattice site. Further, the definition of threshold displacement is explored by making the distinction between atomic displacement ( E ¯ d disp ) and defect formation ( E ¯ d def ). Predicted values of E ¯ d def compare favourably to experimental observations for tungsten and tungsten carbide. Results confirm that E ¯ d def and E ¯ d disp are strongly structure dependent. Differences between E ¯ d disp and E ¯ d def are commensurate with rapid defect recombination within the timeframe of the simulations for some species and structures but not universally. The probability of displacement and defect formation as a function of primary knock-on energy is also reported. Previously developed models for the average displacement of the primary knock-on atom based on kinetic energy and momentum-dependent drag terms are generally found to provide a useful level of approximation. Anisotropy is investigated and results highlight differences due to structures.

Incorporation and Interaction of Co‐Doped Be and Mg in GaN Grown by Metal‐Organic Chemic Vapor Deposition

Despite recent advances in growth and characterization of GaN:Be, reliable conductive p‐type GaN:Be remains elusive. In this work, GaN is co‐doped with Be and Mg to improve the incorporation and ionization efficiency of both Be and Mg. Be and Mg are found to interact in complex ways rendering GaN:Be,Mg semi‐insulating. Rather than improving Be and Mg incorporation efficiency, there is an apparent mutual inhibition of incorporation when Be and Mg are co‐dopants. Furthermore, photoluminescence measurements indicate that the BeGa acceptor is reduced in particular, relative to the total [Be]. The same effect is not observed for Mg. From this, it is concluded that Mg preferentially incorporates into Ga lattice sites over Be, and excess Be that cannot incorporate into substitutional sites instead occupies interstitial sites. The interstitial Be acts as donor defect, which compensate Mg acceptors. This provides an explanation for the observed reduction in BeGa‐related luminescence intensity (without an associated decrease in [Be]) in GaN:Be with significant [Mg] content.

A Multiscale Simulation on Aluminum Ion Implantation-Induced Defects in 4H-SiC MOSFETs

Aluminum (Al) ion implantation is one of the most important technologies in SiC device manufacturing processes due to its ability to produce the p-type doping effect, which is essential to building p–n junctions and blocking high voltages. However, besides the doping effect, defects are also probably induced by the implantation. Here, the impacts of Al ion implantation-induced defects on 4H-SiC MOSFET channel transport behaviors are studied using a multiscale simulation flow, including the molecular dynamics (MD) simulation, density functional theory (DFT) calculation, and tight-binding (TB) model-based quantum transport simulation. The simulation results show that an Al ion can not only replace a Si lattice site to realize the p-doping effect, but it can also replace the C lattice site to induce mid-gap trap levels or become an interstitial to induce the n-doping effect. Moreover, the implantation tends to bring additional point defects to the 4H-SiC body region near the Al ions, which will lead to more complicated coupling effects between them, such as degrading the p-type doping effect by trapping free hole carriers and inducing new trap states at the 4H-SiC bandgap. The quantum transport simulations indicate that these coupling effects will impede local electron transports, compensating for the doping effect and increasing the leakage current of the 4H-SiC MOSFET. In this study, the complicated coupling effects between the implanted Al ions and the implantation-induced point defects are revealed, which provides new references for experiments to increase the accepter activation rate and restrain the defect effect in SiC devices.

Study of CO2 Adsorption Properties on the SrTiO3(001) Surface with Ambient Pressure XPS.

The adsorption properties of CO2 on the SrTiO3(001) surface were investigated using ambient pressure X-ray photoelectron spectroscopy under elevated pressure and temperature conditions. On the Nb-doped TiO2-enriched (1 × 1) SrTiO3 surface, CO2 adsorption, i.e., the formation of CO3 surface species, occurs first at the oxygen lattice site under 10-6 mbar CO2 at room temperature. The interaction of CO2 molecules with oxygen vacancies begins when the CO2 pressure increases to 0.25 mbar. The adsorbed CO3 species on the Nb-doped SrTiO3 surface increases continuously as the pressure increases but starts to leave the surface as the surface temperature increases, which occurs at approximately 373 K on the defect-free surface. On the undoped TiO2-enriched (1 × 1) SrTiO3 surface, CO2 adsorption also occurs first at the lattice oxygen sites. Both the doped and undoped SrTiO3 surfaces exhibit an enhancement of the CO3 species with the presence of oxygen vacancies, thus indicating the important role of oxygen vacancies in CO2 dissociation. When OH species are removed from the undoped SrTiO3 surface, the CO3 species begin to form under 10-6 mbar at 573 K, thus indicating the critical role of OH in preventing CO2 adsorption. The observed CO2 adsorption properties of the various SrTiO3 surfaces provide valuable information for designing SrTiO3-based CO2 catalysts.

Study of CO2 Adsorption Properties on the SrTiO3(001) Surface with Ambient Pressure XPS.

The adsorption properties of CO2 on the SrTiO3(001) surface were investigated using ambient pressure X-ray photoelectron spectroscopy under elevated pressure and temperature conditions. On the Nb-doped TiO2-enriched (1 × 1) SrTiO3 surface, CO2 adsorption, i.e., the formation of CO3 surface species, occurs first at the oxygen lattice site under 10-6 mbar CO2 at room temperature. The interaction of CO2 molecules with oxygen vacancies begins when the CO2 pressure increases to 0.25 mbar. The adsorbed CO3 species on the Nb-doped SrTiO3 surface increases continuously as the pressure increases but starts to leave the surface as the surface temperature increases, which occurs at approximately 373 K on the defect-free surface. On the undoped TiO2-enriched (1 × 1) SrTiO3 surface, CO2 adsorption also occurs first at the lattice oxygen sites. Both the doped and undoped SrTiO3 surfaces exhibit an enhancement of the CO3 species with the presence of oxygen vacancies, thus indicating the important role of oxygen vacancies in CO2 dissociation. When OH species are removed from the undoped SrTiO3 surface, the CO3 species begin to form under 10-6 mbar at 573 K, thus indicating the critical role of OH in preventing CO2 adsorption. The observed CO2 adsorption properties of the various SrTiO3 surfaces provide valuable information for designing SrTiO3-based CO2 catalysts.

Competition between Cubic and Tetragonal Phase Induced Unusual Vertical Magnetization Shift and Exchange Bias in CoxMn3−xO4 (1 ≤ x ≤ 2) Spinel Nanoparticles

CoxMn3−xO4 (1 ≤ x ≤ 2) spinel nanoparticles synthesized through conventional coprecipitation technique exhibit the coexistence of tetragonal and cubic phases in the range of 1 ≤ x ≤ 1.5, while a pure cubic phase is observed for 1.75 ≤ x ≤ 2 at room temperature. Reduction in tetragonal phase fraction from 92 % for x = 1 to 47% for x = 1.5 is attributed to diminution of Jahn–Teller (J–T) active Mn3+ ions occupying the octahedral site of spinel lattice. X‐ray photoelectron spectroscopy confirms both +2 and +3 oxidation states for Co and Mn. Surprisingly, cubic and tetragonal phases exhibit magnetic transition, Tc1 and Tc2, corresponding to a paramagnetic to a high and low temperature ferrimagnetic state, respectively. Tetragonal phase induces high spontaneous (HSEB) and conventional (HCEB) exchange bias with unusually high vertical magnetization shift (VMS) than that of the pure cubic phase, shows maximum HCEB of 4.062 kOe for x = 1.5 and a VMS of 2.5 emu g−1 for x = 1. Such dependence of VMS and exchange bias on tetragonal to cubic phase ratio in CoxMn3−xO4 is demonstrated for the first time and interpreted based on the interaction between different arrangement of spins in tetrahedral and octahedral sublattice.

Cluster Odd-Parity Multipoles by Staggered Orbital Ordering in Locally Noncentrosymmetric Crystals

Odd-parity multipoles in crystals manifest themselves not only in their peculiar electronic orderings but also in unconventional parity-violating physical phenomena. We here report the emergence of odd-parity multipoles by considering staggered orbital orderings in a locally noncentrosymmetric crystal system with the global inversion center but without the inversion center at each lattice site. We show that various odd-parity multipoles, such as the electric toroidal monopole, electric dipole, and electric toroidal quadrupole, are realized depending on the type of orbital orderings in the one-dimensional zigzag chain. Such odd-parity multipoles give rise to an antisymmetric spin splitting in the electronic band structure with the aid of the relativistic spin–orbit coupling. We also show that similar states with odd-parity multipoles are realized in other locally noncentrosymmetric crystals, such as the two-dimensional honeycomb and three-dimensional diamond structures.

Effect of thermal fluctuations on the nontrivial topology of thed + idsuperconducting phase.

The behavior of the topological index (TI), characterizing the properties of superconducting phases of quasi-two-dimensional systems with nontrivial topology, is investigated depending on the temperature and system parameters. For this purpose, a method of calculating the TI, based on a self-consistent functional-integral theory, is proposed. The chirald + idsuperconducting phase of a quasi-two-dimensional model with effective attraction between the electrons located at the nearest sites of a triangular lattice is considered. It is shown that the structure of the energy dependence of the self-energy function, which occurs when taking into account thermal fluctuations, does not lead to a change in the topological properties of the system. It is found that taking into account thermal fluctuations with an increase in the effective attraction between electrons expands the temperature range in which the value of the TI is close to the integerC1≃-2.

Compositional-Dependent Structural Parameters and Energetics of Hibonite: Quantum-Chemical Investigation.

One of the first minerals to condense from the early solar nebula was hibonite (CaAl12O19), demonstrating a hexagonal crystal structure. There are five unique aluminum cation sites (M1-M5) in hibonite. Although hibonite contains aluminum, it can also have 3d transitional metals that substitute at aluminum cation sites, such as titanium, and other metals such as magnesium. Titanium can occur in multiple oxidation states, linking to redox conditions. A single substitution occurs when a single aluminum atom is replaced by a single titanium atom, causing the Ti to take on a 3+ oxidation state. On the other hand, Ti achieves an oxidation state of 4+ during a double substitution when a pair of aluminum atoms is replaced by a titanium and a magnesium atom. A density functional theory-based calculation is used to explore the relationship between the number of substitutions and the lattice parameters of the hibonite crystal. The calculated results reveal that the lengths of both unit cells (a and c) increase with an increasing number of single or double substitutions. The rate of increment per substitution depends on the cation sites, M1-M5, and substitution types. Along with the expansion of the unit cell of the hibonite due to the substitution, the substituted unit cell gains energy (and in turn stability) compared to the substituted unit cell without expansion, dependent on the lattice site and substitutions.

Effect of high pressure synthesis conditions on the formation of high entropy oxides

High entropy materials are often entropy stabilized, meaning that the configurational entropy from multiple elements sharing a single lattice site stabilizes the structure. In this work, we study how high-pressure synthesis conditions can stabilize or destabilize a high entropy oxide (HEO). We study the high-pressure and high-temperature phase equilibria of two well-known families of HEOs: the rock salt structured compound (Mg,Co,Ni,Cu,Zn)O, including some cation substitutions, and the spinel structured compound (Cr,Mn,Fe,Co,Ni)3O4. Syntheses were performed at various temperatures, pressures, and oxygen activity levels, resulting in dramatically different synthesis outcomes. In particular, in the rock salt HEO, we observe the competing tenorite and wurtzite phases and the possible formation of a layered rock salt phase while the spinel HEO is highly susceptible to partial decomposition into a mixture of rock salt and corundum phases. At the highest tested pressures, 15 GPa, we discover the transformation of the spinel HEO into a metastable modified ludwigite-type structure with the nominal formula (Cr,Mn,Fe,Co,Ni)4O5. The relationship between the synthesis conditions and the final reaction product is not straightforward. Nonetheless, we conclude that high-pressure conditions provide an important opportunity to synthesize high entropy phases that cannot be formed any other way.

Deconfined quantum criticality of nodal d -wave superconductivity, Néel order, and charge order on the square lattice at half-filling

We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the π-flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background π-flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal d-wave superconductor, and states with Néel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases that are very likely described by (2+1)-dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and Néel order, and a conventional d-wave superconductor with gapless Bogoliubov quasiparticles at four nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of Néel, valence bond solid (Kekulé), and Dirac semimetal phases. Published by the American Physical Society 2024

Optical and EPR spectroscopy of manganese doped LiNaGe4O9 single crystals

The paper presents data on the study of optical absorption, photoluminescence and EPR of lithium-sodium tetragermanate crystals (LiNaGe4O9) doped with Mn ions. Optical spectra were measured in the range 11000–45000 cm−1 and temperatures 77–300 K. They are attributed to manganese in the +4 charge state. From the analysis of optical spectra, the crystal field strength Dq = 2175.2 cm−1 and Racah parameters A = 4065 cm−1, B = 813 cm−1, C = 3239 cm−1 were determined for the Mn4+ ions. The value of the Huang-Rhys factor SFHR = 17.56 indicates a strong electron-phonon interaction of the Mn4+ ion. The presence of Mn4+ ions was further confirmed by measurement of EPR spectra at microwave frequencies 8.9 and 320 GHz. EPR data show that Mn4+ ions occupy two structurally nonequivalent lattice sites with the formation of two Mn4+ centers: Mn1 and Mn2. From the angular dependences of the EPR spectra, the components of the g factor and the hyperfine interaction tensor A, the parameters of the axial D and rhombic E crystal field were determined for both centers. The axial constant D is large for both centers: 0.644 and 0.560 cm−1, respectively. Analysis of optical and EPR spectroscopic data testifies that Mn4+ ions replace the Ge4+ ions in oxygen octahedrons within the host. Structurally nonequivalent centers are attributed to manganese ions located in undistorted lattice sites (Mn1) and in sites perturbed by the LiNa antisite defect (Mn2).

Predictive Approach to the Phase Behavior of Polymer–Water–Surfactant–Electrolyte Systems Using a Pseudosolvent Concept

A predictive approach to the phase behavior of four-component polymer–water–surfactant–electrolyte systems is formulated by viewing the four-component system as a binary polymer–pseudosolvent system, with the pseudosolvent representing water, surfactant, and the electrolyte. The phase stability of this binary system is examined using the framework of the lattice fluid model of Sanchez and Lacombe. In the lattice fluid model, a pure component is represented by three equation-of-state parameters: the hard-core volume of a lattice site (v*), the number of lattice sites occupied by the component (r), and its characteristic energy (ε*). We introduce the extra-thermodynamic postulate that r and v* for the pseudosolvent are the same as for water and all surfactant–electrolyte composition-dependent characteristics of the pseudosolvent can be represented solely through its characteristic energy parameter. The key implication of the postulate is that the phase behavior of polymer–pseudosolvent systems will be identical for all pseudosolvents with equal values of characteristic energy, despite their varying real compositions. Based on the pseudosolvent model, illustrative phase diagrams have been computed for several four-component systems containing alkyl sulfonate/sulfate surfactants, electrolytes, and anionic or nonionic polymers. The pseudosolvent model is shown to describe all important trends in experimentally observed phase behavior pertaining to polymer and surfactant molecular characteristics. Most importantly, the pseudosolvent model allows one to construct a priori phase diagrams for any polymer–surfactant–electrolyte system, knowing just one experimental composition data for a system at the phase boundary, using available thermodynamic data on surfactants and electrolytes and without requiring any information on the polymer.

Emergent Physics in Metal−Organic Frameworks

AbstractMany‐body interactions in condensed matter could lead to emergent phenomena spanning superconductivity, ferromagnetism, exciton condensation, etc. The emergence of these phenomena often requires highly ordered spatial arrangements of the interacting species to enforce specific space symmetries and interacting strengths. Metal−organic frameworks (MOFs), crystalline materials formed by self‐assembly of metal ions and organic ligands, allow for precise design of their crystal structures and sophisticated tuning of Coulombic interaction or magnetic coupling among lattice sites. Such atomic‐level designability combined with high crystallinity and versatile types of lattices (e.g., kagome and honeycomb lattices) render MOFs as a great platform to investigate emergent physics. In this Emerging Topic, we summarize recent studies evidencing emergent phenomena in MOFs including strong correlations, superconductivity, charge density wave, long‐range magnetic order, and quantum spin liquid. We highlight the great potential of MOFs as quantum materials and discuss challenges including growth of high‐quality single crystals and in‐depth physical characterizations to reveal insights into the nature of physical properties of MOFs. Key Scientists

Observation of reentrant metal-insulator transition in a random-dimer disordered SSH lattice

The interrelationship between localization, quantum transport, and disorder has remained a fascinating focus in scientific research. Traditionally, it has been widely accepted in the physics community that in one-dimensional systems, as disorder increases, localization intensifies, triggering a metal-insulator transition. However, a recent theoretical investigation [Phys. Rev. Lett. 126, 106803] has revealed that the interplay between dimerization and disorder leads to a reentrant localization transition, constituting a remarkable theoretical advancement in the field. Here, we present the first experimental observation of reentrant localization using an experimentally friendly model, a photonic SSH lattice with random-dimer disorder, achieved by incrementally adjusting synthetic potentials. In the presence of correlated on-site potentials, certain eigenstates exhibit extended behavior following the localization transition as the disorder continues to increase. We directly probe the wave function in disordered lattices by exciting specific lattice sites and recording the light distribution. This reentrant phenomenon is further verified by observing an anomalous peak in the normalized participation ratio. Our study enriches the understanding of transport in disordered mediums and accentuates the substantial potential of integrated photonics for the simulation of intricate condensed matter physics phenomena.

Adjusting luminescence properties of ZnAl2O4:Mn2+(Mn4+), Li+ phosphors through cation substitution.

ZnAl2O4 with a typical spinel structure is highly expected to be a novel rare-earth-free ion-activated oxide phosphor with red emission, which holds high actual meaning for advancing phosphor-converted light-emitting diode (pc-LED) lighting. Among the rare-earth-free activators, Mn4+ ions have emerged as one of the most promising activators. Considering the price advantage of MnCO3 generating Mn2+ ions and the charge compensation effect potentially obtaining Mn4+ ions from Mn2+ ions, this research delves into a collection of ZnAl2O4:Mn2+(Mn4+), x Li+ (x = 0%-40%) phosphors with Li+ as co-dopant and MnCO3 as Mn2+ dopant source prepared by a high temperature solid-state reaction method. The lattice structure was investigated using X-ray diffraction (XRD), photoluminescence (PL), and photoluminescence excitation (PLE) spectroscopy. Results suggest a relatively high probability of Li+ ions occupying Zn2+ lattice sites. Furthermore, Li+ ion doping was assuredly found to facilitate the oxidization of Mn2+ to Mn4+, leading to a shift of luminescence peak from 516 to 656 nm. An intriguing phenomenon that the emission color changed with the Li+ doping content was also observed. Meanwhile, the luminescence intensity and quantum yield (QY) at different temperatures, as well as the relevant thermal quenching mechanism, were determined and elucidated detailedly.

Lattice imperfections and high-harmonic generation in correlated systems

We study effects of lattice imperfections on high-harmonic generation from correlated systems using the Fermi–Hubbard model. We simulate such imperfections by randomly modifying the chemical potential across the individual lattice sites. We control the degree of electron–electron interaction by varying the Hubbard U. In the limit of vanishing U, this approach results in Anderson localization. For nonvanishing U, we rationalize the spectral observations in terms of qualitative k-space and real-space pictures. When the interaction and imperfection terms are of comparable magnitude, they may balance each other out, causing Bloch-like transitions. If the terms differ significantly, each electron transition requires a relatively large amount of energy and the current is reduced. We find that imperfections result in increased high-harmonic gain. The spectral gain is mainly in high harmonic orders for low U and low orders for high U.

Highly dispersed spherical phosphors obtained by encapsulating silica hollow spheres with SnO2: Eu3+ layers for enhancing red light emission

Orange light (593 nm, 5D0→7F1) and red light (613 nm, 5D0→7F2) are the two most important emission lines belonging to Eu3+ ions in high and low symmetries, respectively, making them important indicators for judging the nanoenvironment around Eu3+ ions. Therefore, obtaining significant red light emission in Eu3+ doped SnO2 nanocrystals requires anchoring Eu3+ ions onto lattice sites with low symmetries, which can be achieved through maximizing surface doping. To achieve this goal, we coated very thin Eu3+ doped SnO2 nanocrystal (10–20 nm) coatings on extremely dispersed SiO2 hollow spheres (500 nm in diameter) to investigate doping dependent luminescent properties. A significant increase in red emission was observed in such a system, accompanied by a significant attenuation of orange light, and the intensity of red emission increases with increasing Eu3+ concentration. When the concentration of Eu3+ exceeds 3 at%, the intensity of red light emission sharply decreases due to the quenching effect. For the optimal doped hollow sphere sample (3 at%), the asymmetry ratio (red/orange light integration intensity ratio) under indirect excitation is 5.58, which further rises to 11.3 under direct excitation, while the asymmetry ratio of SnO2 nanopowder with the same doping concentration is less than 0.2 under both excitation modes, indicating that the current structural design plays a significant role in the modulation of Eu3+ ion luminescence. Our research not only demonstrates an effective method to enhance the red light emission of Eu3+ doped SnO2 nanocrystals, but also provides a method for manipulating the polychromatic luminescence of such systems.

Formation of Oxygen Vacancies in Cr3+-Doped Hydroxyapatite Nanofibers and Their Role in Generating Paramagnetism

AbstractWe demonstrate that doping hydroxyapatite (HAp) with Cr3+ ions induces oxygen vacancies, contributing to paramagnetism. Cathodoluminescence and photoluminescence analyses reveal increased oxygen vacancy formation in $${\text{O}}{\text{H}}^{-}$$ OH - and $${\text{P}}{\text{O}}_{4}^{3-}$$ PO 4 3 - groups with rising Cr3+ concentrations, highlighted by stronger cathodoluminescence emissions at 2.57 and 2.95 eV and the photoluminescence emission at 3.32 eV. Raman spectroscopy shows new modes at 900 and 970 cm−1, indicating distortion of the v1 vibrational mode due to Cr3+ substitution at Ca(II) sites of the HAp lattice. X-ray photoelectron spectroscopy confirms Cr3+ in the HAp:Cr. Magnetometry reveals a shift from diamagnetism in pure HAp to increasing paramagnetism in HAp:Cr with higher Cr3+ content, achieving 0.0460 emu/g at 10 kOe with concentrations higher than 2.9 at.%. This paramagnetism is attributed to Cr3+ ions and singly ionized oxygen vacancies $$V^{\prime}_{{\text{O}}}$$ V O ′ aligning along an external magnetic field, with $$V^{\prime}_{{\text{O}}}$$ V O ′ formation linked to $${\text{PO}}_{4}^{{3}-}$$ PO 4 3 - replacement by $${\text{PO}}_{3}^{{2}-}$$ PO 3 2 - in HAp.

The Dynamics of the Hubbard Model Through Stochastic Calculus and Girsanov Transformation

As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=UN$$\\end{document} fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the difference of expected number of particles at the two lattice sites is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.