In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.
Exact Sum Rules for Vector Channel at Finite Temperature and its Applications...Daisuke Satow
Slides used in presentation at:
“International School of Nuclear Physics 38th Course Nuclear matter under extreme conditions -Relativistic heavy-ion collisions”, in September, 2016 @ Erice, Italy
In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.
Exact Sum Rules for Vector Channel at Finite Temperature and its Applications...Daisuke Satow
Slides used in presentation at:
“International School of Nuclear Physics 38th Course Nuclear matter under extreme conditions -Relativistic heavy-ion collisions”, in September, 2016 @ Erice, Italy
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Speedlancer is the world's fastest freelance marketplace - Agency designers, delivering work in just 4 hours.
Some of our design portfolio as of August 2015
Want to hire us? Get in touch at support [at] speedlancer.com! Or order now at http://Speedlancer.com and have your design back in just 4 hours!
Also offering content writing & data-entry services.
Neutral Electronic Excitations: a Many-body approach to the optical absorptio...Claudio Attaccalite
Neutral Electronic Excitations: a Many-body approach to the optical absorption spectra.
Introduction to Bethe-Salpeter equation and linear response theory.
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Speedlancer is the world's fastest freelance marketplace - Agency designers, delivering work in just 4 hours.
Some of our design portfolio as of August 2015
Want to hire us? Get in touch at support [at] speedlancer.com! Or order now at http://Speedlancer.com and have your design back in just 4 hours!
Also offering content writing & data-entry services.
10 Steps to the Perfect Interim Hire - Network Executiveaglass3
Interim management has rapidly risen in popularity in recent years to become one of the most highly
favoured strategic resources to effect fast change and progress when organisations need it most...
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Quantum-Gravity Thermodynamics, Incorporating the Theory of Exactly Soluble Active Stochastic Processes, with Applications
by Daley, K.
Published in IJTP in 2009. http://adsabs.harvard.edu/abs/2009IJTP..tmp...67D
Lecture by prof. dr Neven Bilic from the Ruđer Bošković Institute (Zagreb, Croatia) at the Faculty of Science and Mathematics (Niš, Serbia) on October 29, 2014.
The visit took place in the frame of the ICTP – SEENET-MTP project PRJ-09 “Cosmology and Strings”.
Classical and Quasi-Classical Consideration of Charged Particles in Coulomb F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb centres is carried out. It is shown that the proton and electron can to create a stable connection with the dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom coincide with the Born orbits.
CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the
Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity
theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb
centres is carried out. It is shown that the proton and electron can to create a stable connection with the
dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the
proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of
the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with
the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy
for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom
coincide with the Born orbits.
CLASSICAL AND QUASI-CLASSICAL CONSIDERATION OF CHARGED PARTICLES IN COULOMB F...ijrap
On the basis of the theory of bound charges the calculation of the motion of the charged particle at the
Coulomb field formed with the spherical source of bound charges is carried out. Such motion is possible in
the Riemanniam space-time. The comparison with the general relativity theory (GRT) and special relativity
theory (SRT) results in the Schwarzshil'd field when the particle falls on the Schwarzshil'd and Coulomb
centres is carried out. It is shown that the proton and electron can to create a stable connection with the
dimensions of the order of the classic electron radius. The perihelion shift of the electron orbit in the
proton field is calculated. This shift is five times greater than in SRT and when corrsponding substitution of
the constants it is 5/6 from GRT. By means of the quantization of adiabatic invariants in accordance with
the method closed to the Bohr and Sommerfeld one without the Dirac equation the addition to the energy
for the fine level splitting is obtained. It is shown that the Caplan's stable orbits in the hydrogen atom
coincide with the Born orbits.
Poster of my master\'s research presented at the Physics@FOM conference at Veldhoven on 20 januari 2010. There\'s one error in the equations, can you find it?
1 ECE 6340 Fall 2013 Homework 8 Assignment.docxjoyjonna282
1
ECE 6340
Fall 2013
Homework 8
Assignment: Please do Probs. 1-9 and 13 from the set below.
1) In dynamics, we have the equation
E j Aω= − −∇Φ .
(a) Show that in statics, the scalar potential function Φ can be interpreted as a voltage
function. That is, show that in statics
( ) ( )
B
AB
A
V E dr A B≡ ⋅ = Φ −Φ∫ .
(b) Next, explain why this equation is not true (in general) in dynamics.
(c) Explain why the voltage drop (defined as the line integral of the electric field, as
defined above) depends on the path from A to B in dynamics, using Faraday’s law.
(d) Does the right-hand side of the above equation (the difference in the potential
function) depend on the path, in dynamics?
Hint: Note that, according to calculus, for any function ψ we have
dr dx dy dz d
x y z
ψ ψ ψ
ψ ψ
∂ ∂ ∂
∇ ⋅ = + + =
∂ ∂ ∂
.
2) Starting with Maxwell’s equations, show that the electric field radiated by an impressed
current density source J i in an infinite homogeneous region satisfies the equation
( )2 2 iE k E E j Jωµ∇ + = ∇ ∇⋅ + .
Then use Ampere’s law (or, if you prefer, the continuity equation and the electric Gauss
law) to show that this equation may be written as
( )2 2 1 i iE k E J j J
j
ωµ
σ ωε
∇ + = − ∇ ∇⋅ +
+
.
2
Note that the total current density is the sum of the impressed current density and the
conduction current density, the latter obeying Ohm’s law (J c = σE).
Explain why this equation for the electric field would be harder to solve than the equation
that was derived in class for the magnetic vector potential.
3) Show that magnetic field radiated by an impressed current density source satisfies the
equation
2 2 iH k H J∇ + = −∇× .
Explain why this equation for the magnetic field would be harder to solve than the
equation that was derived in class for the magnetic vector potential.
4) Show that in a homogenous region of space the scalar electric potential satisfies the
equation
2 2
i
v
c
k
ρ
ε
∇ Φ + Φ = − ,
where ivρ is the impressed (source) charge density, which is the charge density that goes
along with the impressed current density, being related by
i ivJ jωρ∇⋅ = −
Hint: Start with E j Aω= − −∇Φ and take the divergence of both sides. Also, take the
divergence of both sides of Ampere’s law and use the continuity equation for the
impressed current (given above) to show that
1 ii v
c c
E J
j
ρ
ωε ε
∇⋅ = − ∇⋅ = .
Note: It is also true from the electric Gauss law that
vE
ρ
ε
∇⋅ = ,
but we prefer to have only an impressed (source) charge density on the right-hand side of
the equation for the potential Φ. In the time-harmonic steady state, assuming a
homogeneous and isotropic region, it follows that ρv = ρvi. That is, there is no charge
3
density arising from the conduction current. (If there were no impressed current sources,
the total charge density would therefore be ze ...
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
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This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
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zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex Proofs
E05731721
1. IOSR Journal of Engineering (IOSRJEN) www.iosrjen.org
ISSN (e): 2250-3021, ISSN (p): 2278-8719
Vol. 05, Issue 07 (July. 2015), ||V3|| PP 17-21
International organization of Scientific Research 17 | P a g e
Bose-Einstein Condensation Controlled by (HOP+OLP) Trapping
Potentials
Noori.H.N. Al-Hashimi andEman A.M. AL-Malki
Department of Physics; College of Education for pure science,University of Basra; Basra; Iraq
Abstract:-This work will focus on theoretical treatment of one-dimension harmonic oscillator trapping potential
HOP applied along the x-axis together with an optical lattice trapping potential OLP applied along y-axis.
These trapping potentialsare usually used in experiments that lead to produced Bose-Einstein condensation BEC
in ultra-cold gases. This analysis will concentration mainly on the anisotropy parameter in HOPand q parameter
in OLP, and the role that these factors play in term of values and shapes of the trapping potential, initial wave
function, and final wave function. This specific study will give us the overall view of the region of confinement
that the HOP and OLPhave employed. The study is extended to the chemical potential of the working fluid.
Keywords: Laser cooled atom, BEC atom, Trapping, Quantum Oscillator
I. INTRODUCTION
A periodic potential can be designed simplyby overlapping two counter-propagating laser beams [1].
The interference between the two laser beams forms an optical standing wave with period λL/2, which can trap
the atoms. By interfering more laser beams, onecan obtain one, two, and three-dimensional (1D, 2D and 3D)
periodic potentials [2-5]. The 1D lattice, formed by a pair of laser beams, generates a single standing wave
interference pattern excellently an array of 2D disk-like trapping potentials [6,7]. Two orthogonal optical
standing waves can create an array of 1D potential tube, in which the atoms can only move along the weakly
confining axis of the potential tube, thus grasping 1D quantum performance, with the radial motion being
completely frozen out for low-enough temperatures. Three orthogonal optical standing waves correspond to a
3D simple cubic crystal, in which each trapping site acts as a tightly confining harmonic oscillator potential.
One important advantage of using optical fieldsto create a periodic trapping potential is that the geometry and
depth of the potential are under acomplete control of the experimentalist. For example,the geometry of the
trapping potentials can be alteredby interfering laser beams under a different angle, thus making even more
complex lattice arrangements [8], such as Kagome lattices [9]. The deepness of such optical potentials can even
be varied dynamically during an experimental sequence by simply increasing or decreasing the intensity of the
laser light, thus turning experimental investigations of the time dynamics of essential phase transitions into a
reality.As a simulator, an optical lattice offersremarkably clean access to a particular Hamiltonianand thereby
assists as a model system for testing fundamental theoretical concepts, at times providing good examples of
quantum many-body effects.Usually, ultra-cold neutral atoms are put in storage inmagnetic traps, in which only
a small subset of the available atomic spin states, the so-called weak field-seeking states can be trapped. This
restrictionis generally overcome by using optical dipole traps that rely on the interaction between an induced
dipole moment in an atom and an external electric field. Such a field can, for example, be provided bythe
oscillating electric light field from a laser, which induces an oscillating dipole moment in the atom while at the
same time interacts with this dipole moment in order to create a trapping potentialVdip(r) for the atoms [10]. In
this study we will focus on the relations that connect the chemical potential and wave function with the
anisotropy of the HOP, and the chemical potential with the parameter q of the OLP. The study is extended to
analysis the distribution of the wave function through the working area. We have confirmed that the parameters
in HOP, and OLP play a major part in the solution of the GPE and care should be taking into account if one like
to bring his analysis to a satisfactory experiments result.
II. THEORY
The Gross-Pitaevskii Equation (GPE), is the result of a mean-fieldtheory that’s most useful in
treatinghighly dilute and ultra-cold BECs withnegligible many-body correlation effects.It has also been useful in
dealing with physicalsituations in which a cloud of thermally excitedbosonic atoms or fermionic atoms
accompany agaseous condensate. In these cases, the GPE’s numericalsolution must be found in conjunction
withthe solution of appropriate equations for the coexistingcloud, such as the Vlasov-Landau equations [11].
Under these conditions it is worth first to write down the GPE. If we consider the macroscopic dynamics of a
BECloaded onto an elongated optical lattice, similar to thosecreated in a number of experiments (see, e.g.,
Refs.[12-14]). The condensate dynamics is described by thethree-dimensional GP equation,
2. Bose-Einstein Condensation Controlled by (HOP+OLP) Trapping Potentials
International organization of Scientific Research 18 | P a g e
iℏ
∂ψ
∂t
= −
ℏ2
2m
∇2
+ V x, y, z + Γ ψ 2
ψ (1)
Where ψ(x, y, z, t) is the wave function of the cigar-shapedcondensate, (y,z) are the directions of
strongtransverse confinement, and y is the direction of thelattice. The combined potential of the optical
latticeand magnetic trap V(x, y,z) can be written as,
V y, r2
= E0sin2 πy
d
+
1
2
m(ωx
2
x2
+ ω⊥
2
r2
) (2)
E0 is the well depth of the opticallattice, d is the characteristic lattice constant, and ωiare trapping frequencies in
the corresponding directions.The parameter Γ = 4πℊℏ2
as/m characterizes the s-wavescattering of atoms in the
condensate which introducesan effective nonlinearity in the mean-field equation; it ispositive for repulsive
interactions and negative for attractiveinteractions.Measuring time in units of ω⊥
−1
, the spatial variablesin units
of the transverse harmonic oscillator length,a0 = ℏ/mω⊥
1/2
, the wave function amplitude in unitsof a0
−3/2
,
and the potential in units of ℏω⊥, we obtain thefollowing dimensionless GP equation:
i
∂ψ
∂t
= −
1
2
∇2
+ V x, y + G2d ψ 2
ψ (3)
Where,t = Tω⊥, and G2d = 4π as/a0 . The potential now takes the form:
V x, y = V0sin2
qy +
1
2
AL 2
x2
(4)
Where, V0 = E0/ ℏω⊥ ,and q = πa0/d. The ratio of the confinement for the magnetic trapAL = ωx/ω⊥varies
from 10−1
to,1/ 2, see for example [15-18], but this is not the case in this study.The Crank-Nicolson (CN)
scheme is the slogger for the numerical integration of quantum wave equations. The method’s major advantage
is that it ensures unitarily for an arbitrary size of the time step dt. This lets the wave function advance by large
steps—that is, larger than the inverse of the highest eigen value,ωMAX = Emax ℏ. Only accuracy (at the second
order) limits the time-step size, not stability or unitarily considerations. The CN method requires us to solve a
matrix algebraic problem at each time step. This is a fairly expensive computational task.
To solve the GPE several properties should be assign that are:
a) A first property of the GPE is the time invariance. This means that the equation is time reversible: the
equation is stable with respect to the time change of variable t → −t.
b) A second property is known as invariance by phase shift or gauge invariance: if ψ is the solution to equation
(3), then ψ = ψe−αt
is solution to (3) with,V → V + α.
c) Invariants: mass conservation The mass is defined by:
N(t) = N(ψ(t,·)) = |ψ|2
x
dx, ∀t > 0. (5)
Then we can prove that we have the mass conservation N(t) = N(0).
d) Another conserved quantity is the energy: let us introduce the energy functional Eψ as
Eψ = {
1
2
ψ 2
+ V(x)x
ψ 2
+ β( ψ 2
)}dx, ∀t > 0. (6)
where the primitive Β of β is defined as:
Β λ = β λ dλ
λ
0
. (7)
Then we have the energy conservation property,Eψ = Eψo
.
e) A last property of interest is related to the dispersion relation providing hence special solutions: the plane
wave
ψ(t, x) = aei(kx −ωt)
(8)
is solution to system (3) (with V = 0) if the following nonlinear dispersion relation holds
ω = ||k||2
/ 2 + β(|a|2
), (9)
where || · || is the usual 2-norm for complex valued vector fields.
By taking all these into account the numerical solution of the equation (3) will take the form:
−1
ψi,j
∗
− ψi,j
n
k
= −
1
8
ψi+1,j
∗
− ψi,j
∗
+ ψi−1,j
∗
+ ψi+1,j
n
− ψi,j
n
+ ψi−1,j
n
+
ψi,j+1
∗
− ψi,j
∗
+ ψi,j−1
∗
+ ψi,j+1
n
− ψi,j
n
+ ψi,j−1
n
+
3. Bose-Einstein Condensation Controlled by (HOP+OLP) Trapping Potentials
International organization of Scientific Research 19 | P a g e
Vi,j
2
ψi,j
∗
+ ψi,j
n
+ G2d
ψi,j
n
2
2
ψi,j
∗
+ ψi,j
n
(10)
III. RESULT AND DISCUSSION
In this study, we examine and extend the analysis of Bose-Einstein condensation in an external
harmonic potential applied along X-axis, and optical lattice potentials applied along Y-axis. The two-
dimensional system of N non-interacting Bosons with the particles contained in an isotropic two-dimensional
harmonic potential and optical lattice potential are analysis in this work. In this paper we have examined several
features of BEC in harmonic potentials and optical lattice potential for the ideal gas theoretically which we are
going to explain it as follow: First of all it's worth to mention that the working field is divide into 1000 point in
space coordinates, and divide into 300000 point in time coordinates, the space interval is DX = 0.020, DY =
0.020 and the time interval is DT = 0.0001, the nonlinearity is fixed to the value of 12.548, the total number of
runs for each case is 500 and the total number of pass is 3000. Under these conditions we can examine several
cases along the direction of understanding of BEC as follows:
1) Figure (1) shows the relation of the chemical potential with the parameter q that appears in optical lattice
potential. This parameter is responsible on the distribution of the optical lattice potential along Y-axis. The
oscillator harmonic potential for this case kept constant with the value of anisotropy equal to 1.0. one can
conclude from this figure that the value of the optical lattice potential play an important role in determining
the chemical potential of the system. The relation of the chemical potential (µ) of the system with the optical
lattice potential in term of parameter q can fit the polynomials:
μ q = −1.432 ∗ 10−8
q6
+ 4.121 ∗ 10−6
q5
− 4.177 ∗ 10−4
q4
+ 0.0180q3
− 0.291q2
+ 3.127q − 8.355.
2) Figure (2) shows the wave function as a function of an isotropy (AL) for fixed value of optical lattice
potentials. From this figure one can says that there is a logarithm relation between the wave function and
the an isotropy of the harmonic oscillator potential and this relation can be written as:
ψ(AL) = 0.1591 ln AL + 0.2103, 0.5 ≤ Al ≤ 7.0
It is worth to say that this relation is restricted to the values of the anisotropy with the range 0.5 ≤ Al ≤ 7.0
3) Figure (3) shows the chemical potential as a function of an isotropy for fixed values of optical lattice
potential. This figure is true for the range of anisotropy 0.5 ≤ Al ≤ 7.0, and the relation between the
chemical potential and the an isotropy can fit a second order polynomial which is look like the one given
below:
μ AL = 0.24994806 Al 2
+ 0.00025534148 AL + 70.638367, 0.5 ≤ Al ≤ 7.0
4) Figure (4) shows a two dimension distribution of wave function on the XY-plane. This distribution is
calculated for fix value of the optical lattice potential in term of the dimensionless parameter q equal to 45,
and for two difference values of harmonic oscillator potential in term of the dimensionless an isotropy
values which are 0.5, and 6.0 respectively. One can say from this figure that the wave function distribution
of the condense gas are increases remarkable as the anisotropy increase. The transvers distribution of the
wave function along the y-axis is almost controlled by the OPL only, and since the OPL is constant along
this axis so, we can notice from this figure that the condense gas distribution along the transvers axis is not
affected at all, although the values of the wave function are not the same.
5) In order to analysis a comprehensive view of the distribution of wave function, one have to compare
between the initial and final distributions of the wave functions for a fixed values of the trapping potentials
along the X-axis, and Y-axis as seen in figures (5). Figure (5a) shows the 2D distributions of wave function
which is initially in the ground state of the trap potential and is therefore stationary. The effect of an optical
lattice trapping potentials on this distribution is clear from this figure. The wave functionis separating into a
series of localized pieces as shows in figure (5a), with the contour map of the distribution of wave function
as its projected in XY plane. The potential energy operator therefore takes the form V x, y = V0sin2
qy +
1
2
AL 2
x2
, where ALis the trap constant, qis the laser field wave number, V0 is the lattice intensity. In
applications to quantum computing, this localized wave function is to represent quantum bits. However, due
to the nonlinearity of the equations, the condensate develops a phase that varies from lattice site to lattice
site as seen in this figure, which is undesirable for quantum computing, since these algorithms assume that
there are zero relative phases among the various single quantum bits. The problem is therefore to eliminate
this phase profile by adjusting the trap strength by applying the laser field in term of the optical lattice
potentials. The trap constant AL is taken as the control and the objective is to minimize the variance of the
phase of the wave function. In doing that and by keeping the nonlinearity of the equations under control,
4. Bose-Einstein Condensation Controlled by (HOP+OLP) Trapping Potentials
International organization of Scientific Research 20 | P a g e
and after 500 runs with 2000 pass of the computer programs the shape of the distribution of final wave
function is behave very well as expected and the phase became constant from lattice site to lattice site as
one can see this clear in figure (5b). This figure shows a 2D view of the distribution of final wave function
together with a contour projection of this distribution on the XY-plane. As a conclusion one can say that the
optical lattice trapping potential together with the oscillator harmonic potential play an important part on the
distribution of the initial and final wave functions of the condensation and by a carful adjustment of these
trapping potential on can bring the calculated values to a real experiment results.
5. Bose-Einstein Condensation Controlled by (HOP+OLP) Trapping Potentials
International organization of Scientific Research 21 | P a g e
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