Bose-Einstein Condensation and Quantized Vortices in Superfluidity and Superconductivity
(1 Nov - 31 Dec 2007)
~ Abstracts ~
Quantum phase transition and
transport of cold atoms in an optical lattice
Wuming Liu, Chinese Academy of Sciences, China
We first study the superfluid–Mott-insulator phase
transition of ultracold dilute gas of bosonic atoms in an
optical lattice by means of Green function method and
Bogliubov transformation as well. The superfluid–Mott-insulator
phase transition condition is determined by the energy-band
structure with an obvious interpretation of the transition
mechanism. Moreover the superfluid phase is explained
explicitly from the energy spectrum derived in terms of
Bogliubov approach [1].
The superfluid–Mott-insulator phase transition of dipolar
bosons in optical lattice is analyzed. By using the
Bogoliubov approach and decoupling approximation, the energy
spectrum and zero-temperature phase diagram of dipolar
bosonic atoms in an optical lattice are obtained
analytically. The results show that in these systems the
superfluid–Mott-insulator phase transition can be induced by
tuning the dipole-dipole interaction and the position of the
phase boundary can be moved when the on-site interaction is
varied. Corresponding to a large on-site interaction, the
dipole-dipole interaction is attractive near the critical
point of the superfluid–Mott-insulator phase transition [2].
The exact macroscopic wave functions of two-species
Bose-Einstein condensates in an optical lattice beyond the
tight-binding approximation are studied by solving the
coupled nonlinear Schrodinger equations. The phase diagram
for superfluid and insulator phases of the condensates is
determined analytically according to the macroscopic wave
functions of the condensates, which are seen to be traveling
matter waves [3].
We also propose a scheme to investigate the magnetic phase
transition of cold atoms confined in an optical lattice. We
also demonstrate how to get coupled two-leg spin ladder
which display a phase transition from spin liquid to
magnetic ordered state in two-dimensional optical lattice.
An experimental protocol is further designed for observing
this new phenomenon [4].
We design a quantum transport device -- Josephson atomic
quantum dot device, which is an atomic quantum dot coupled
to two superfluid Bose-Einstein condensate reservoirs via
Raman transition. Our results show that the resonant peak of
the transmission probability becomes wide as the Rabi
frequency increases, which is insensitive to the interaction
parameter in the reservoirs because of the strong
collisional interaction in the dot. The Josephson current
reaches the maximum and appears a \pi phase shift as Raman
detuning approaching to the resonant energy. We also
demonstrate that the Josephson atomic quantum-dot can be
also regarded as a superfluid quantum interference device.
How to achieve the parameter region of this quantum device
is discussed in present experimental condition [5].
References
[1] J. J. Liang, J. Q. Liang, W. M. Liu, Quantum phase
transition of condensed bosons in optical lattices, Phys.
Rev. A68, 043605 (2003).
[2] Z.W. Xie, W. M. Liu, Superfluid-Mott-insulator
transition of dipolar bosons in optical lattice, Phys. Rev.
A 70, 045602 (2004).
[3] G. P. Zheng, J. Q. Liang, W. M. Liu, Phase diagram of
two-species Bose-Einstein condensates in an optical lattice,
Phys. Rev. A 71, 053608 (2005).
[4] P.B. He, Q. Sun, P. Li, S.Q. Shen, W.M. Liu, submitted
Phys. Rev. A.
[5] J. M. Wang, J.Q. Liang, W.M. Liu, submitted Phys. Rev.
Lett.
Interacting bosons under
confinement
Mukunda P Das, The Australian National University,
Australia
The successful observation of Bose-Einstein condensation
(BEC) in weakly interacting dilute alkali gases, confined by
inertial magneto-optic traps, has provoked a flood of
activities both theoretically and experimentally. Recently
ground state properties of interacting and confined Bosons
are mainly studied by the stationary solutions of Gross-Pitaevskii
(GP) mean-field equation. In this talk I shall present a
general microscopic theory of confined interacting Bosons at
non-zero temperature. The method employed for this
investigation is the double time temperature Green function
technique. I shall show some new results beyond the usual
mean-field theory and highlight the crucial role of
temperature, confinement and interaction to the properties;
namely energetics and depletion of the condensates. I shall
also discuss the collective modes of the confined Bosons.
Vortex matter phase transition
in type II superconductors
Mukunda P Das, The Australian National University,
Australia
The magnetic phase diagram for a normal type II
superconductor has several important features, most notable
of which are the lower and upper critical fields. Between
the lower and upper critical fields the magnetic field lines
penetrate the superconductor in the form of discrete field
quantum, known as a vortex. As the magnetic field increases,
more of field lines penetrate the superconductor, so the
density of the vortices increases. The vortices organise in
order to minimise the total free energy what is called an
Abrikosov vortex lattice. The lattice spacing depends on the
temperature T and applied magnetic field H. If there is a
lattice at T≠0, it has to thermally melt undergoing a solid
to a fluid phase transition.
In high temperature cuprate superconductors the phase
diagram of (H~T) is far more complex. It has been widely
explored yielding a rich variety of new physics. In this
talk I shall discuss a microscopic approach to study
freezing of the vortices of a fluid to a vortex solid phase.
One will note that this is opposite to melting. The method I
shall employ is the density functional theory of freezing.
Applied magnetic field fixes the areal density of the
vortices and I shall require the intra- and inter-pair
potentials of the vortices. Some details of calculated and
experimental phase diagrams will be presented.
Quantum world of ultra-cold
atoms
Christopher Foot, University of Oxford, UK
Nowadays it is possible to cool atoms to temperatures
less than a millionth of a degree (microkelvin) above
absolute zero and this enables us to study the many
fascinating quantum mechanical properties of atomic systems
at such extremely low temperatures. The lecture will
describe the tremendous advances in physics that have made
such experiments possible, and which led to the Nobel prizes
in physics for the “development of methods to cool and trap
atoms with laser light” in 1997, and for the “achievement of
Bose-Einstein condensation in dilute gases of alkali atoms”
in 2001. It seems counterintuitive that shining laser light
on atoms cools them and this will be explained, together
with the way that laser beams are used to hold the cold
atoms at fixed positions in space and arrange them into
regular patterns to construct ultra-cold quantum matter.
The concepts will be explained without mathematics in a
manner suitable for a general audience.
Matter-wave gap solitons and
vortices in optical lattices
Yuri S. Kivshar, Australian National University,
Australia
We overview our recent theoretical results on the study
of the existence, stability, and generation of spatially
localized matter-wave gap solitons [1], self-trapped
nonlinear states [2], and vortices [3] in Bose-Einstein
condensates with repulsive atomic interactions confined by
one- and two-dimensional optical lattices. Akin to bright
gap solitons in optics, these nonlinear localized states
exist only within the gaps of the matter-wave bandgap
spectrum imposed by the periodicity of the lattice
potential. We discuss the complex structure of matter-wave
phase singularities associated with different types of
stationary gap vortices and suggest two different excitation
methods. We demonstrate that the broad vortices are
intimately connected to self-trapped nonlinear states [2]
recently demonstrated in experiments with one-dimensional
optical lattices.
Our numerical simulations also confirm the feasibility of a
homodyne interferometric detection of broad gap vortices.
[1] E.A. Ostrovskaya and Yu.S. Kivshar, Opt. Express 12, 19
(2004).
[2] T.J. Alexander, E.A. Ostrovskaya, and Yu.S. Kivshar,
Phys. Rev. Lett.
96, 040401 (2006).
[2] T.J. Alexander, E.A. Ostrovskaya, and Yu.S. Kivshar,
Phys. Rev. A 74, 040401, 023605 (2006).
Rapidly rotating Bose-Einstein
condensates
Alexander L. Fetter, Stanford University, USA
Trapped Bose-Einstein condensates (BECs) differ
considerably from the standard textbook example of a uniform
Bose gas. In an isotropic harmonic potential, the
single-particle ground state introduces a new intrinsic
scale of length (the ground-state siz) and energy (the
ground-state energy). When the trap rotates at low angular
velocity,
the behavior of a single vortex illustrates the crucial role
of discrete quantized vorticity. For more rapid rotation,
the condensate contains a vortex array. The resulting
centrifugal forces expand the condensate radially and shrink
it axially; thus the condensate becomes effectively
two-dimensional. If the external rotation speed approaches
the frequency of the radial harmonic confining potential,
the condensate enters the ``lowest-Landau-level" regime, and
a simple description again becomes possible. Eventually, the
system is predicted to make a quantum phase transition to a
highly correlated state analogous to the fractional quantum
Hall states of electrons in a strong magnetic field.
Strong confinement for the
Gross Pitaevski equation : energy solutions
Naoufel Ben Abdallah,Toulouse University and CNRS,
France
The limit of the Gross Pitaevski equation under partial
strong confinement leads to a infinite system of coupled
nonlinear Schrodinger equations in the non confined
dirtection. In a previous work (cf. the talk of F. Mehats),
smooth local in time solutions are proven to exist for the
limiting system and convergence is addressed.
The keypoint is to rewrite the limiting system as a unique
three dimensional Schrodinger equation with a nonlinearity
where oscillations due to the confining potential are
filtered out.
The aim of the present talk is to prove existence and
uniqueness of energy solutions for the limiting equation and
to address convergence issues in this framework.
We first show that the averaging formulation holds for
energy solutions, then prove galigriardo-Nirenberg
inequalities for the averaged nonlinearity and finally use
nonisotropic Strichartz estimates to prove crucial L
infinity bounds on the wavefunction.
Nonlinear coherent destruction
of tunneling
Biao Wu, Chinese Academy of Sciences, China
We study a two-mode nonlinear model under a periodic
driving, which can be used to describe a BEC in a
double-well potential under a periodic driving. With a
self-developed numerical method, we are able to compute all
its Floquet states and the corresponding quasi-energies.
With these results, we argue that the localization
phenomenon presented in this nonlinear driving system is
closely related to the coherent destruction of tunneling
(CDT) in a linear driving system. Therefore, we call it
nonlinear coherent destruction of tunneling (NCDT). In
contrast
to that CDT occurs only at isolated parameter points, NCDT
happens for a wide range of parameters. NCDT may also be
observable with Bose-Einstein condensate. However, we find
that a better system to observe NCDT is two coupled
periodically-curved optical waveguides with Kerr
nonlinearity.
Mathematical studies for elliptic
systems arising from BEC
Juncheng Wei, The Chinese University of Hong Kong,
Hong Kong
We discuss variousmathematical questions for elliptic
systems modelling BEC. These include: existence and
uniqueness of ground states and bound states, the effect of
trapping potentials on the existence o bound states, a
priori estimates, phase separations, existence of skymions,
etc.
Existence of stationary
solutions to the exterior problem for the Boltzmann equation
Tong Yang, City University of Hong Kong, Hong Kong
The exterior problem arising in the study of the flow
past an obstacle is one of the most classical and important
subjects in gas dynamics and fluid mechanics. We consider
the problem when the bulk velocity at infinity is assigned
which is not a trivial driving force on the flow. The result
to be presented generalizes the previous one on the same
problem by Ukai-Asano to more general boundary conditions by
crucially using the velocity average argument.
Vortices in quantum condensed
phases. The case of spin 0 and of spin 1/ 2 particles.
Yves Pomeau, University of Arizona, USA
Based on a very general argument on the symmetry of the
quantum ground state of a system of particles, Onsager
showed the possibility of vortices as topologically stable
defects connected to the U(1) symmetry of the ground state
linked to phase rotations, this being true for spin zero
particles. In a system of particles with spin 1/2, the
ground state has now the symmetry SU(2)xU(1) and topological
defects may include also a continuous rotation of all spins.
I shall write the mechanical equations valid whenever the
parameters of the SU(2)xU(1) symmetry change slowly in space
and time. I will show also that this leads very naturally to
divide by a factor of two of the quantum of London in
superconductors, as observed, independent on any particular
assumption on the structure of this ground state besides its
SU(2)xU(1) symmetry.
Dipolar quantum gases: bosons and
fermions
Han Pu, Rice University, USA
Compared to their counterparts with
short-range interactions, quantum gases with long-range
interactions behave differently in many qualitative ways.
One important long-range interaction is the dipolar
interaction between either magnetic or electric dipoles of
atoms or molecules. Dipolar effects in Bose-Einstein
condensate has recently been unambiguously observed in
chromium condensate. There are however many interesting
phenomena remain to be explored. In this talk, I will
discuss dipolar effects in both BEC and quantum degenerate
fermions.
BEC: ratcheting, scattering, and
entanglement
Sergej Flach, Max-Planck-Institut fuer Physik
komplexer Systeme, Germany
Hamiltonian ratchets harvest on broken space-time
symmetries. Recent experiments with thermal cold atoms in
optical potentials resulted in a beautiful confirmation of
the theoretical concept. The single particle quantum regime
allows to resonantly enhance the ratchet current. I will
introduce these concepts, and present recent studies of the
influence of atom-atom interactions in a BEC (on the GP
level) on the ratchet current and resonances. Spatially
confined atom-atom interactions allow to scatter matter
waves by a localized BEC and to observe destructive
intereference in the two channels of the Boloyubov-de Genne
equations. Going beyond the mean field GP level, I will
finally show, that localized states of few interacting atoms
persist solely due to the presence of entanglement.
Incompressible and compressible
limits of two-component Gross-Pitaevskii equations with
rotating fields and trap potentials
Tai-Chia Lin, National Taiwan University, Taiwan
Recently, a rich variety of dynamical phenomena and a
turbulent relaxation have been observed in rotating
Bose-Einstein condensates depicted by Gross-Pitaevskii
equations coupled with rotating fields and trap potentials.
The dynamical phenomena range from shock-wave formation to
anisotropic sound propagation. The turbulent relaxation
leads to the crystallization of vortex lattices. To see the
dynamical phenomena and the turbulent relaxation of
two-component rotating Bose-Einstein condensates, we study
the incompressible and the compressible limits of
two-component systems of Gross-Pitaevskii equations.
Nonlocal heat flows and a
partition problem for Eigenvalues
Fanghua Lin, New York University, USA
TBA.
« Back...
Numerical continuation for
computing energy levels of Bose-Einstein condensation
Cheng-Sheng Chien, National Chung-Hsing University,
Taiwan
We study some continuation algorithms for computing
energy levels of Bose-Einstein condensation (BEC). First we
transform the nonlinear Schrödinger equation (NLS) to a
nonlinear eigenvalue problem by using a well-known formula
of separation of variables. Then we compute the first few
energy levels of the associated Schrödinger eigenvalue
problem (SEP). Various discretization methods are described
to discretize the Laplacian. The proposed algorithm has the
advantage that it is unnecessary to discretize or integrate
the partial derivatives of wave functions. Moreover, the
wave functions can be computed for any time scale. Numerical
results on the ground state solutions of the BEC, rotating
BEC and BEC with periodic potentials are reported.
Asymptotics for nonlinear
Schrödinger equations with periodic potentials
Christof Sparber, University of Vienna, Austria
We review recent results on rigorous asymptotic studies
of nonlinear Schrödinger equations with rapidly oscillating
periodic potentials. These models naturally appear in the
description of so-called lattice Bose-Einstein condensates.
We will also show recent numerical simulations of such
problems, based on Bloch-decomposition pseudo-spectral
codes.
A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations
Xavier Antoine, Institut Elie Cartan Nancy, France, Institut National Polytechnique de Lorraine, France
In this talk we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present and compare several approaches to implement the classical transparent boundary condition into finite difference and finite element discretizations. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case. This is a joint work with A. Arnold (Vienna University of Technology), C. Besse (University of Lille), M. Ehrhardt (Technische Universitat Berlin) and A. Schädle (Konrad-Zuse-Zentrum fur Informationstechnik Berlin).
Bose-Einstein condensates and optical lattices
Dieter Jaksch, Oxford University, UK
The achievement of ultracold temperatures in weakly interacting atomic gases has led to a range of novel physical setups for studying many body quantum physics. In this talk I will discuss the basic physics of ultracold degenerate quantum gases. I will start by explaining the physics of ideal degenerate quantum gases. This will be followed by a brief discussion of experimental methods for reaching quantum degeneracy in atomic gases. I will then show how weak interactions between the atoms can be accounted for and give an overview of the most important properties of weakly interacting Bose-Einstein condensates.
I will then turn to the case of Bose-Einstein condensates loaded into optical lattices. In particular I will discuss under which conditions the assumption of weak interactions breaks down and show how the system reaches the strongly interacting limit. Finally I will describe current experiments demonstrating strong correlations ultracold atomic samples and studying their striking features.
Polarons in optical lattice immersions
Dieter Jaksch, Oxford University, UK
Ultracold atoms trapped in optical lattices provide an ideal test bed for investigating the physics of strongly correlated quantum systems. They possess several distinguishing features like versatility of setup and large degree of controllability of the system parameters.
Therefore optical lattices can mimic a range of condensed matter systems displaying strong correlations and furthermore realize exotic quantum phases not available otherwise. In addition they are promising candidates for demonstrating quantum information processing applications.
In my talk I will present our recent studies on the dynamics of optical lattices which are immersed in a degenerate quantum gas. The interaction between lattice atoms and background gas leads to the emission of phonons which introduces an adjustable degree of decoherence in the lattice dynamics. Furthermore it results in longer range coherent interactions between the lattice atoms. I will show how the emission of phonons into the background gas gives rise to the formation of polarons [1]. These polarons tunnel through the lattice coherently or incoherently depending on the system parameters. Furthermore I will discuss long range polaron-polaron interactions, the resulting formation of polaron clusters and their stability [2]. I will show that these phenomena are accessible using current technology.
[1] M. Bruderer, A. Klein, C.R. Clark, and D. Jaksch, Phys. Rev. A 76,
011605(R) (2007).
[2] A. Klein, M. Bruderer, S.R. Clark and D. Jaksch, Dynamics and clustering of impurity atoms in Bose-Einstein condensates, preprint.
Mathematical analysis and numerical simulation for Bose-Einstein condensation
Weizhu Bao, National University of Singapore
In this 6-hours tutorial talk, I will review the mathematical results of the dynamcis of Bose-Einstein condensate (BEC) and present some efficient and stable numerical methods for computing ground states and dynamics of BEC. As preparatory steps, we take the three-dimensional (3D) Gross-Pitaevskii equation (GPE), scale it to obtain a three-parameter model and show how to reduce it to 2D and 1D GPE in certain limiting parameter regimes. Then we study numerically and asymptotically the ground states, excited states and quantized vortex states as well as their energy and chemical potential diagram in BEC. Different numerical methods for computing ground states of BEC are reviewed and compared. Some very interesting numerical results of ground and excited states in BEC are reported. Several dynamical laws including conservation of angular momentum expectation, dynamics of condensate width and analytical solutions of stationary states with a shift are derived rigoriously. Different numerical methods for computing the dynamics of BEC are reviewed and compared. The mathematical results and numerical methods for single component nonrotating BEC are extended to rotating BEC and multi-component BEC as well as spin-1 BEC. Finally, different models for BEC at finite temperature are discussed.
Efficient and accurae numerical methods for simulating Bose-Einstein condensates
Weizhu Bao, National University of Singapore
In this talk, I will review different efficient and accurate numerical methods for computing ground states and dynamics of Bose-Einstein condensates (BEC). Ground states and dynamics of BEC in one-dimension (1D), 2D and 3D with different external potentials are reported to demonstrate our numerical methods. The methods are also applied to the collapse and explosion of BEC and numerical results are closed compared with experimental results. Finally, these numerical methods for one-component nonrotating BEC are extended to rotating BEC, multi-component BEC and spin-1 BEC.
Josephson vortex dynamics in stacked intrinsic Josephson junctions with rectangular hole arrays
Beiyi Zhu, Chinese Academy of Sciences, China
The dynamics of Josephson vortices play important roles in superconductor electronics. Especially the coherent Josephson vortex motion in layered high-Tc superconductors, such as Bi_2Sr_2CaCu_2O_8(BSCCO), may lead to the electromagnetic wave emission up to terahertz frequencies due to the strong coupling between the alternating superconducting and insulating layers with atom thickness. The high frequency oscillations can be excited inside the intrinsic Josephson junctions(IJJ) in BSCCO with a finite bias current along the c-axis under in-plane magnetic fields. However, it turns out that synchronization of the Josephson vortex motion is essential for coherent emission. Here, we report our recent study of the Josephson vortex dynamics in the IJJ stack with rectangular hole arrays including experimental measurements and numerical simulations. The periodic mesh structure in the IJJ stack presents significant forced coherent oscillation of the Josephson vortex motion, which are expected to be useful for designing practical devices. This is a joint work with H. B. Wang(National Institute for Materials Science, Japan).
On some special solutions and regimes of the Gross-Pitaevskii equation
Fabrice Bethuel, University of Paris VI, France
The Gross-Pitaevskii equation exhibits a large variety of regimes and special solutions. In this talk, I will focus first on the existence and properties of traveling wave solutions in dimensions 1,2 and 3 (this part of the talk is based on recent joint work with Philippe Gravejat and Jean-Claude Saut). I will stress in particular the transsonic KP-I limit for small energy traveling waves in dimension 2.
In a seconf part, I will discuss another, though related regime, which, from a certain point of view, might be considered as a long wave length limit of the problem (this part is based on a joint ongoing work with Raphaël Danchin and Didier Smets).
Experiments on the superfluid properties of BEC
Christopher Foot, University of Oxford, UK
We have trapped a Bose-Einstein condensed gas of rubidium atoms in a toroidal trapping potential created by a combination of a magnetostatic quadrupole field and a radio-frequency field. This geometry is suitable for studying persistent currents and superfluidity in systems where the dimensionality is 3, 2 or 1. In the talk, I will also describe progress towards loading ultra-cold atoms into rotating optical lattices (which is related to theoretical work carried out in Oxford by Dr Dieter Jaksch), and give a summary of previous experiments to probe the superfluid properties of BEC including observation of the scissors mode and nucleation of vortices.
Semi-classical dynamics in schroedinger equations: convergence and computation
Hailiang Liu, Iowa State University, USA
This talk starts with a nonlinear Schroedinger equation with rotating forcing, a fundamental mean field model for Bose-Einstein condensates.
Using a modified WKB approach, we present a rigorous semi-classical analysis for underlying solutions. This yields a rigorous justification for the hydro-dynamical system of rotating super-fluids. In particular, it is shown that global-in-time semi-classical convergence holds whenever the limiting hydro-dynamical system has global smooth solutions.
Semi-classical convergence becomes more subtle if the hydro-dynamical system admits finite time singularity. We review recently developed level set methods for capturing semi-classical limits in Schroedinger equations with different potentials. We discuss the essential ideas behind the techniques, the coupling of these techniques to handle several canonical potentials, including the phase space based level set method for given potentials; the field space based level set method for self-consistent potentials; as well as the Bloch-band based level set method for periodic potentials. The relations between computed multi-valued solutions and desired physical observables are established. One and two-dimensional numerical results will be presented.
This talk is based on some recent works joint with Christof Sparber (University of Vienna) and Zhongming Wang (Iowa State University).
Hyperplane-constrained continuation method for coupled nonlinear schrodinger equations
Weichung Wang, National Taiwan University, Taiwan
Time-independent m-coupled nonlinear Schrodinger equations (NLSEs) are studied analytically and numerically. Starting from a one-component discrete nonlinear Schrodinger equation (DNLSE), we first propose and analyze an iterative method for finding the ground state solution. This solution is then used as the initial point of the primal stalk solution curve of the m-coupled DNLSEs in a continuation method framework. To overcome the stability and efficiency problems arising in standard continuation methods, we propose a hyperplane-constrained continuation method by adding additional constraints while following the solution curves. Furthermore, we analyze solution and bifurcation properties of the primal stalk solution curve corresponding to the 3-coupled DNLSEs. We also demonstrate computational positive bound states and bifurcation diagrams of the 3-coupled DNLSEs, including non-radially symmetric ground states that are tricky to find in NLSEs.
This is joint work with Yuen-Cheng Kuo, Wen-Wei Lin, and Shih-Feng Shieh.
Efficient and stable spectral methods for unbounded domains:
applications to Bose-Einstein condensates
Jie Shen, Purdue University, USA
We shall review some recent results on numerical analysis
of Laguerre and Hermite spectral methods as well as rational
spectral methods for solving PDEs on unbounded domains. We
shall also present some applications of these methods to the
Gross-Pitaevskii euqation for Bose-Einstein Condensates.
Numerical method for computing ground states of spin-1 Bose-Einstein condensate
Fong Yin Lim, National University of Singapore
The imaginary time method and normalized gradient flow have been widely used to solve the Gross-Pitaevskii equation for the ground states of Bose-Einstein condensates (BEC). In this talk, an accurate and efficient numerical method, the backward Euler sine-pseudospectral (BESP) method, will be presented to discretize the normalized gradient flow and solve for the single component BEC ground state. For spinor condensate with internal degree of freedom, difficulties arise if the imaginary time method is applied directly as there are insufficient normalization conditions. For the reason, a normalization condition is introduced to spin-1 condensate, in addition to the two existing conditions: the conservation of total particle numbers and the conservation of magnetization. The third normalization condition is derived from the relations between the chemical potentials of each spinor component together with a splitting scheme applied to the continuous normalized gradient flow. The BESP scheme is then extended to compute the ground state of spin-1 BEC condensate, which is described by the three-component coupled Gross-Pitaevskii equations.
A variety of shock wave problems in Bose-Einstein condensates
Mark Hoefer, National Institute of Standards and Technology, USA
Recent experiments have demonstrated the existence of blast waves in Bose-Einstein condensates (BECs), eg. steep, propagating, oscillatory fronts [1]. In this context, the Gross-Pitaevskii equation that models the mean field of a BEC is considered in the zero-dispersion
limit. A dispersive regularization known as Whitham averaging will be
used to show that these steep fronts can be considered dispersive
shock waves (DSWs). Using this regularization technique, several BEC
DSW problems will be discussed including the "shock tube" [1] and "piston shock" [2] problems and shock wave interactions [3]. Comparison of the
asymptotic theory with numerical simulation shows quantitative agreement.
Krylov deferred corrections, method of lines transpose, and fast elliptic solvers for time dependent partial differential equations
Jingfang Huang, University of North Carolina, USA
In this talk, we discuss a new class of numerical methods for the accurate and efficient integration of time dependent partial differential equations.
Unlike traditional method of lines (MoL), the new Krylov deferred correction
(KDC) accelerated method of lines transpose (Mol^T) first discretizes the temporal direction using Gaussian type nodes and spectral integration, and the resulting coupled elliptic system is solved iteratively using Newton-Krylov techniques such as Newton-GMRES method, in which each function evaluation is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated MoL^T technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time step sizes in long-time simulations. Numerical experiments for parabolic type equations including the Schrodinger equation will be discussed.
Towards a Bose Einstein condensate on a high temperature superconducting atom chip
Rainer Dumke, Nanyang Technological University, Singapore
Superconductor technology applied for atom optical systems will launch a new generation of compact and reliable atom chips. This approach opens the possibility to scale, parallelize, and miniaturize atom optics for new investigations in fundamental research and potential applications and even pair quantum solid state devices with neutral atoms on the quantum mechanical level. It will lead to new, compact sources of ultra cold atoms, compact sensors based on matter wave interference and new approaches towards quantum computing with neutral atoms and solid state devices. The exploitation of the unique features of the quantum mechanical behavior of matter waves and the capabilities of powerful state-of-the-art fabrication techniques to miniaturize high temperature superconductor devices lend this approach a significant advancement. During the talk I will give a brief overview about the experimental techniques to realize a Bose Einstein condensate on a superconducting atom chip and discuss the current status of the experiment which is still in the initial starting up phase.
Enhanced sensitivity to fundamental constants in ultracold atoms and molecules near feshbach resonances
Cheng Chin, University of Chicago, USA
Scattering length, which can be measured in Bose-Einstein condensate and Feshbach molecule experiments, is extremely sensitive to the variation of fundamental constants, in particular, the electron-to-proton mass ratio m_e/m_p or m_e/lambda, where lambda is the QCD scale. Based on
single- and two-channel scattering models, we show how the variation of the mass ratio propagates to the scattering length. Our results suggest that variation of m_e/m_p on the level of 10^-11 ~ 10^-14 can be detected near a narrow Feshbach resonance by monitoring the scattering length on the 1 % level. In this talk, I will present evidences that demonstrate the ultrahigh high sensitivity on atomic mass and suggest possible experiment approaches to precisely determine the scattering lengths.
Symmetry breaking in spinor-dipolar Bose-Einstein condensates
Masahito Ueda, Tokyo Institute of Technology, Japan
Emergence of order caused by spontaneous symmetry breaking is one of the most important keys to our understanding of macroscopic quantum phenomena. Investigating this problem can explain why mean-field theories have been so successful in elucidating the nature of gaseous BEC systems and when many-body effects play a signigicant role. In this talk I shall demonstrate that gaseous BECs offer a new paradigm for studying the problem of emergence of order by focusing on four distinct cases: namely, soliton formation in a scalar BEC, where the translational symmetry is broken [1]; vortex nucleation in a rotating BEC, where the axisymmetry is broken [2]; spontaneous magnetization in a spinor BEC, where the rotational and chiral symmetries are broken [3]; and spontaneous formation of spin textures in a dipolar BEC, where we can expect the Einstein-de Haas effect [4] and the ground-state mass flow [5]. Possible experimental conditions to observe these phenomena will also be discussed [6].
[1] R. Kanamoto et al., Phys. Rev. Lett. 94 (2005) 090404.
[2] M. Ueda and T. Nakajima., Phys. Rev. A 73 (2006) 043603.
[3] H. Saito et al., Phys. Rev. Lett. 96 (2006) 065302.
[4] Y. Kawaguchi et al., Phys. Rev. Lett. 96 (2006) 080405.
[5] Y. Kawaguchi et al., Phys. Rev. Lett. 97 (2006) 130404.
[6] Y. Kawaguchi et al., Phys. Rev. Lett. 98 (2007) 110406.
Nodal sets and singular sets of solutions to magnetic Schrodinger equations in three dimensions
Xingbin Pan, East China Normal University, China
In this talk we shall discuss some of recent progress on the nodal sets of solutions of the equations involving the magnetic Schrodinger operator. The study of nodal sets of solutions to such type equations has been motivated by the mathematical theory of superconductivity and liquid crystals. It is well- known that the complexity of the nodal set of a function mainly comes from the singular set on which both the function and the gradient vanish. The singular set of a real-valued solution of a linear elliptic equation has been well investigated. For a complex-valued solution of a linear equation involving a magnetic Schrodinger operator, the structure of the nodal set has not been well investigated yet except in the two-dimensional case. In this talk we shall show that the singular set of such a solution in a three-dimensional domain is countably 1-rectifiable. The functions considered in this paper include the order parameter in the Ginzburg-Landau theory of superconductivity, and the eigenfunctions of the magnetic Schrodinger operator. In the talk we shall also discuss some questions related to the nucleation of zeros of the solutions of Ginzburg-Landau equations.
Point vortex dynamics and Kelvin-Helmholtz instability
Jian-Guo Liu, University of Maryland, USA
Point vortex dynamics system can be used effectively as a computational method for singular (fluid as well super-fluidity fluid) flows with its dynamics behavior dominated by vortices concentrated on a low dimension set. These kind of low dimensional concentration of the vortices such as vortex sheets, usually leads to an ill-posed mathematical problem which exhibits the Kelvin-Helmholtz instability.
This class of computational method is known as vortex method, first proposed by Rosehead in1932 for vortex sheet problem, and generalized by A.
Chorin in 70's into
a blob vortex method as a successful computational method for general incompressible flow. Much of mathematical study for this problem has been conducted by
G. Birkhoff, T. Beale, A. Majda, G. Cottet, R. Caflish, R. Kransy,
J. Goodman, T. Hou, J. Lowengrub etc.
In this talk, we will show that Rosehead's original point vortex method for the vortex sheet problem indeed converges, beyond the formulation of singularity, to a classic weak solution of the Euler equation. This is a joint work with Z. Xin.
On dimension reduction of the Gross Pitaevskii equation
Peter Markowich, University of Cambridge, UK and University of Vienna, Austria
We analyse dimension reduction techniques for the Gross- Pitaevskii equation. Typically they are based on steep confining potentials in certain directions. Analytical and numerical results are presented.
Sharp stability of attractive Bose-Einstein condensates
Jian Zhang, Sichuan Normal University, China
We propose the critical nonlinear SchrÄodinger equation
with a harmonic potential as a model of attractive Bose-Einstein condensates. By an elaborate mathematical analysis we show that a sharp
stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental
data. Moreover with this threshold we prove that a ground state of the
condensate exists and is orbital stable. We also evaluate the minmum
of the condensate energy.
Ratchet dynamics of solitons in Bose-Einstein condensates
Dario Poletti, National University of Singapore
We study the ratchet dynamics of bright solitons in Bose-Einstein condensates. In the regime of small perturbations the current is higly affected by the number of atoms in the soliton, so that the average velocity depends on the effective mass of the solitons. We employ this feature to study collisions between ratchet-driven solitons of different masses. Soliton collisions display repulsion independently of the initial phase of the soliton. We show that transport can be induced through interaction between solitons.
Vortices in superconductors
Francois Peeters, Universiteit Antwerpen, Belgium
The physics of vortices in classical fluids and optics has been a very important subject for more than a century both for fundamental science and for applications. Half a century ago, vortices became prominent as quantum mechanical objects constructed from a macroscopic wavefunction as found in superconductivity, superfluidity and more recently in Bose-Einstein condensates. Over the last decade a new field appeared: ‘vortex electronics’ which is nowadays also called ‘fluxonics’
Mesoscopic superconductors
Superconductivity is a macroscopic phenomena and I will discuss how this quantum state is influenced if the sample is reduced in size.
In recent years a large variety of different geometries of mesoscopic superconductors have been investigated, e.g. disks, rings, squares, triangles, double ring structures, etc. When they are placed in an external magnetic field a rich variety of novel vortex configurations were found, e.g. giant vortex, ring configurations of vortices, combinations of giant and multi-vortex configurations, vortex-antivortex pairs, etc.
It was found that the vortex state of a mesoscopic superconductor is strongly determined by its size and to a lesser extend by the material parameters the superconductor is made of. For example, by increasing the radius of a flat superconducting disk it is possible to obtain a magnetic response which, as function of the magnetic field, is continuous, type-I, a type-I with multiple steps and type-II. A detailed comparison with experiments will be made.
The exact geometry of the superconductor has also a strong influence on its vortex state. Commensurability effects between the geometry of the boundary of the sample and the triangular vortex lattice may induce new vortex states consisting of an anti-vortex surrounded by vortices. Up to now there is no experimental confirmation of this new vortex state.
The vortex state in a mesoscopic sample can be brought into a very long lived metastable state owing to the presence of barriers for flux penetration and exit. This leads to unexpected effects like fractional flux and even negative flux entry. In bulk superconductors the flux associated to a vortex is quantized. In mesoscopic superconductors the flux is not necessarily quantized. This is even more so in small superconducting rings were several experiments are available.
Vortices in a superconducting disk can be viewed as classical particles which may form ordered structures, i.e. they are localized on rings. These ordered structures can exhibit different metastable configurations like for the case of classical particles. Transitions between such states occur through saddle points in the energy landscape. By increasing the magnetic field sufficiently the ordered structure can melt into a uniform state, i.e. a liquid like state.
The properties of perforated mesoscopic samples, in particular superconducting squares and thin films, with regular patterns of submicron antidots or blind holes will be discussed. Their vortex structure, critical parameters, different commensurate vortex configurations, possible degeneracies and transitions between them will be analysed. In addition, we consider a superconducting square with 2x2 blind holes as an alternative logic device, analogue of the Quantum Cellular Automata (QCA), where quantum dots are replaced by blind holes and vortices take the role of the charged particles. Particular logic state is labeled by the position of two vortices in one of the two diagonal directions, and can be manipulated by the magnetic field of the current loops adequately placed on top of the sample. We demonstrate the possible realization of arrays of such logic cells as QCA circuits, performing given logic operations.
I will discuss the vortex physics in superconducting films containing a lattice of anti-dots.
New vortex configurations are found between the anti-dots: giant-vortices, combination of giant- and multi-vortex states, as well as symmetry imposed vortex-antivortex states for particular geometrical parameters of the sample. The anti-dot occupation number is calculated as a function of relevant parameters and compared with existing expressions for the saturation number and with experimental results. Novel behavior of the vortex anti-dot interaction is predicted for particular sizes of the anti-dot.
For small radius of the anti-dots a triangular vortex lattice can be stabilized where some of the vortices are pinned by the anti-dots and some of them are located between them.
The enhanced critical current at integer and rational matching fields is found, where the level of enhancement at given magnetic field directly depends on the vortex-occupation number of the anti-dots. For certain parameters of the anti-dot lattice and/or temperature the critical current is found to be larger for higher magnetic fields as found recently in experiments.
Hybrid systems
The interplay between superconductivity and the inhomogeneous magnetic field generated by nanostructured ferromagnets leads to new vortex arrangements not found in homogeneous superconductors.
I will consider two situations: 1) a ferromagnetic disk on top of a thin superconducting film, and 2) a lattice of such ferromagnetic disks separated by a thin oxide layer on top of a thin superconducting film.
For a single ferromagnetic disk magnetized perpendicular to the plane of the superconducting film we found that antivortices are stabilized in shells around a central core of vortices (or a giant vortex) with size/magnetization-controlled ‘magic numbers’. The transition between the different vortex phases occurs through the creation of a vortex-antivortex pair under the edge of the magnetic disk.
In the case of a lattice of ferromagnetic disks, the antivortices form a rich spectrum of lattice states. In the ground state the antivortices are arranged in the so-called matching configurations between the ferromagnetic disks while the vortices are pinned to the ferromagnets. The exact (anti)vortex structure depends on the size, thickness and magnetization of the magnetic dots, periodicity of the ferromagnetic lattice and properties of the superconductor expressed through the effective Ginzburg-Landau parameter k*. The experimental implications of our results such as magnetic-field-induced superconductivity will be discussed.
The theoretical analysis is based on a numerical ‘exact’ solution of the phenomenological Ginzburg-Landau equations.
Formalism and numerical techniques
In the London approach the vortices are treated as classical point particles. The vortex configurations are obtained using molecular dynamics simulations (or even with Monte Carlo simulations). The dynamics of vortices is overdamped. Non-linear dynamics and time dependent phenomena can be studied.
A more accurate approach is based on a mean field approach where the vortices are extended objects. This formalism leads to the non-linear Ginzburg-Landau equations. These are two coupled non-linear differential equations that we were able to solve numerically ‘exact’ for thin superconductors of arbitrary shape in the presence of a perpendicular magnetic field. The numerical procedure is based on the finite difference technique.
I will show that in case of samples with high symmetry already qualitative information can be obtained using an expansion of the order parameter in the solutions of the linear Ginzburg-Landau equation.
The time dependent Ginzburg-Landau equation is used to describe the transition between different metastable states.
The three vortex loops state in superconductors with a magnetic core
Francois Peeters, Universiteit Antwerpen, Belgium
In the fifties, Abrikosov found the coexistence of superconductivity with an external magnetic field in the form of vortices, quantized filaments that cross the material. The puzzling properties of the recently discovered ferromagnetic superconductors and superconducting ferromagnets are indicative of vortices, but this time stemming from an internal magnetic field.
Here we analyze this latter phenomenon assuming intrinsic magnetic domains and propose a confined vortex phase, prior to the onset of long range order as a spontaneous vortex phase. Curiously this study bears similarity to the quark confinement problem because of the ’t Hooft and Mandelstam made nearly thirty years ago. They proposed that the duality between electric charges and magnetic monopoles present in Maxwell’s theory is the key to understand quark confinement. Thus instead of trying to understand the confinement of electric charges one should look to the dual problem, the confinement of magnetic monopoles. A pair formed by a monopole and an anti-monopole inside a superconductor is realistic and experimentally feasible because it just corresponds to a tiny magnetic inclusion inside the superconductor. Here we show that a magnetic moment either inside an extreme type II superconductor or in a sandwich structure, such as for a tri-layer superconduc-tor-ferromagnetic-superconductor, gives rise to confined vortex states in its neighbor-hood. For a small magnetic moment there are exactly three confined vortex loops whereas for a large one more elaborate vortex arrangements arise are possible. Their onset from the upper critical field core in sets of threes is shown here to be ener-getically favorable over the growth of just one or two confined vortex loops for different boundary conditions. All our results are derived in the context of the Ginzburg-Landau theory and their validity checked for different choice of boundaries and parameters. We show how the presence of such states can be tracked down in the I-V characteristic curve, thus providing a simple method to their detection. The Lorentz force causes their growth and periodic evolution, passing through a long range ordered state (bulk) or an external vortex pair state (sub-micron superconduc-tor).
Work done in collaboration with: Mauro M. Doria, Antonio R. de C. Romaguera en M. V. Milosevic
New dynamic transition theory and its applications to superconductivity
Shouhong Wang, Indiana University, USA
In this talk, I shall present a new dynamic transition theory for nonlinear phase transitions. The theory consists of 1) a new classification scheme of phase transitions, 2) methods to identify the types of transitions, and 3) dynamic models for equilibrium phase transitions.
Application to superconductivity is based on the time-dependent Ginzburg-Landau model. We demonstrate that there are two type of dynamic transitions, jump and continuous, dictated by the sign of a nondimensional parameter R. This parameter is computable, and depends on the material property, the applied field, and the geometry of domain that the sample occupies. Furthermore, using the parameter R, precise analytical formulas for critical domain size, and for critical magnetic fields are derived. If time permits, the applications to superfluidity will also be addressed.
This is joint work with Tian Ma.
A mass and magnetization conservative and energy diminishing numerical method for computing ground state of spin-1 Bose-Einstein condensates
Hanquan Wang, Yunnan University of Finance and Economics, China
In this talk, a normalization (or mass) and magnetization conservative and energy diminishing numerical method is presented for computing the ground state of spin-1 (or F = 1 spinor) Bose-Einstein condensates (BEC). We begin with the coupled Gross-Pitaevskii equations and the ground state is defined as the minimizer of the energy functional under two constraints on the normalization and magnetization. By constructing a continuous normalized gradient flow (CNGF) which is normalization and magnetization conservative and energy diminishing, the ground state can be computed as the steady state solution of the CNGF. The CNGF is then discretized by Crank-Nicolson finite difference method with a proper way to deal with the nonlinear terms and we prove that the discretization is normalization and magnetization conservative and energy diminishing in discretized level. Numerical results of the ground state and their energy of spin-1 BEC are reported to demonstrate the efficiency of the numerical method.
Superradiance from Bose-Einstein condensate
Xiaoji Zhou, Peking University, China
The dot-like pattern with weak coupling case and line-like pattern with strong coupling are observed in the superradiance experiment in elongated Bose-Einstein condensate where the incident pulse light travels along its long axis. The obvious spatial shape in the weak coupling and the atom number distribution in the high-order mode show the coupling dynamic of the optical and matter wave field, which are the first reported. Our theoretic mode is good agreements with the obtained experimental results which demonstrated that the spatial propagation effects are important to analysis experimental phenomenon. Furthermore, a general semi-classical theory for the superradiance in Bose-Einstein Condensate with two ground states is given, and a good candidate to produce large number of atom-atom entanglement pair is suggested.
Vortex dynamics and interactions in rotating Bose-Einstein condensates
Yanzhi Zhang, Florida State University, USA
One manifestation of superfluidity in Bose-Einstein condensate (BEC) is the appearance of quantized vortices. In this talk, we will numerically and analytically show the phenomena of quantized vortices in rotating BEC. Firstly, we show the rich patterns of the vortex lattices in rotating BEC confined in different external trapping potentials. Then in the weakly interacting BEC, the interactions of two or more vortices having the same or opposite winding numbers are numerically studied. Our numerical results are compared with the analytical results. Finally the dynamics of vortex lattices are also presented.
Joint work with Weizhu Bao, Dieter Jaksch, Alexander Klein and Rong Zeng.
Cold atoms and BECs with time-periodic driving
Tania Monteiro, University College London, UK
I will discuss theoretical and experimental work on the dynamics of cold atoms and BECs subjected to time periodic driving.
At the non-interacting limit, coherent matter wave dynamics of atoms subjected to short pulses ('kicks') from sinusoidal optical lattices provide a rich probe of
quantum chaotic dynamics [1]; new possibilities arise for manipulating cold atoms, leading to a experimental demonstration of a fully coherent quantum ratchet [2]. In the presence of interactions, experiments with kicked BECs mean the dynamical stability of the condensate is now of interest. We investigate the mechanism for destabilization using propagation of the GPE with the time-dependent Bogoliubov approach and show the stability border found for kicked BECs is unrelated to chaos [3].
In certain quantum many- body Hamiltonians, AC-driving offers possibilities for controlling quantum state transfer in spin-Hamiltonians [4] and manipulating the Superfluid-Mott Insulator transition by coherent destruction of tunnelling (CDT) [5].
[1]C.Creffield, G.Hur, T.S. Monteiro, Phys.Rev.Lett., {93} 024103 (2006); W.Jiao, T S Monteiro, S Fishman, J.Keating, R Schubert accepted for PRL, 2007.
[2] Monteiro T. S. et al, Phys.Rev Lett {89} 194102 (2002); P. H. Jones, M. Goonasekera, D. R. Meacher,T. Jonckheere, T. S. Monteiro, Phys.Rev.Lett., {98} 073002 (2007).
[3]J.Reslen, C.E.Creffield, T.S.Monteiro. {\em quant-ph:0707.1653} (2007) [4]T Boness, S Bose, T.S. Monteiro, Phys.Rev.Lett., {96} 187201 (2006) [5] C.E. Creffield and T. S. Monteiro,Phys.Rev.Lett., {96} 210403 (2006)
Resonant Excitations in
Bose-Einstein Condensates
Paulsamy Muruganandam, Bharathidasan University, India
One manifestation of superfluidity in Bose-Einstein condensate (BEC) is the appearance of quantized vortices. In this talk, we will numerically and analytically show the phenomena of quantized vortices in rotating BEC. Firstly, we show the rich patterns of the vortex lattices in rotating BEC confined in different external trapping potentials. Then in the weakly interacting BEC, the interactions of two or more vortices having the same or opposite winding numbers are numerically studied. Our numerical results are compared with the analytical results. Finally the dynamics of vortex lattices are also presented.
Joint work with Weizhu Bao, Dieter Jaksch, Alexander Klein and Rong Zeng.
Graphene
Wuming Liu, Chinese Academy of Sciences, China
Graphene sheets, one-atom-thick two-dimensional layers of
sp$^2$-bonded carbon -- are predicted to have a range of
unusual properties. There are 21 papers published in Nature,
23 papers in Science and 120 papers in Physical Review
Letters since 2005.
In this talk, I will review the recent research results for
its some novel properties such as massless Dirac fermions,
quantum Hall effect and Berry phase in grapheme. I also
introduce our recent results -- the conductivity $\sigma$ of
graphene nanoribbons with zigzag edges as a function of
Fermi energy $E_F$ in the presence of the impurities with
different potential range. The dependence of $\sigma(E_F)$
displays four different types of behavior, classified to
different regimes of length scales decided by the impurity
potential range and its density. Particularly, low density
of long range impurities results in an extremely low
conductance compared to the ballistic value, a linear
dependence of $\sigma(E_F)$ and a wide dip near the Dirac
point, due to the special properties of long range potential
and edge states. These behaviors agree well with the results
from a recent experiment by Miao et al. Science 317, 1530
(2007). We also investigated the effect of topological
defects on the transport properties of a narrow ballistic
ribbon of graphene with zigzag edges. The results show that
the longitudinal conductance vanishes at discrete Fermi
energies where the system develops loop orbital electric
currents with certain chirality. The chirality depends on
the direction of the applied bias voltage and the sign of
curvature created by topological defects. This novel quantum
blockade phenomenon provides a new way to generate magnetic
moment by an external electric field, which can prove useful
in carbon electronics.
Effect of trapping
potential for interacting bosons on optical lattice
Xiaoqun Wang, Renmin University of China, China
Much experimental work has been devoted to detecting a
Mott transition driven by a boson-boson interaction from a
superfluid to a Mott insulator for ultra-cold bosonic atoms
on optical lattice with a trapping potential, but the
intrinsic role of the trapping potential has not received
attention. Density matrix renormalization group studies for
one-dimensional case shows that the trapping potential
essentially introduce an energy scale and can significantly
interplay the competition between a kinetic energy and a
boson-boson interaction, resulting in intriguing phenomenon.
The effect indicates that current experimental observations
of the Mott transition may not match the theoretical one
which presumably mimics the conventional one in condensed
matter physics.
A semiclassical transport model for thin quantum barriers
Shi Jin, University of Wisconsin, USA
We present a time-dependent semiclassical transport model for mixed state scattering with thin quantum barriers.
The idea is to solve a stationary Schrodinger equation in the thin quantum barrier to obtain the scattering coefficients, and then use them to supply the interface condition that connects the two classical domain. We then build in the interface to the numerical flux, in the spirit of the Hamiltonian-preserving scheme introduced by Jin and Wen for a classical barrier. The overall cost is basically the same as solving a classical barrier. We construct a numerical method based on this semiclassical approach, and verify the validity of the model using various numerical examples.
Threshold for global existence of Gross-Pitaevskii equation
Guanggan Chen, Sichuan Normal University, China
We study the Gross-Pitaevskii equation in terms of the partial differential equation theory and variational methods. By establishing various constrained variational problems with invariant evolution flows, we obtain the threshold for global existence and blowup of Gross-Pitaevskii equation.
This is joint work with Jian Zhang.
Interactions and dynamics of quantized vortices
Qiang Du, Pennsylvania State University, USA
The appearance of quantized vortices is a well-known signature of superfluidity. They have been studied extensively in superfluid Helium, type-II superconductors, Bose Einstein condensates and more recently in Fermi gas. We will discuss some mathematical and numerical studies of the quantized vortices. Particular attention will be given to vortex nucleations, interactions between individual vortices and vortex clusters, and the effect on the interactions due to the geometry and topology, as well as external fields such as the applied electric current.
Vortex stability and dynamics in Ginzburg-Landau-Schroedinger
and nonlinear wave equation
Weizhu Bao, National University of Singapore
In this talk, I will review our recent work on quantized vortex stability and dynamics in Ginzburg-Landau-Schroedinger and nonlinear wave equation for modeling superfluidity and superconductivity as well as nonlinear optics. The reduced dynamic laws for quantized vortex interaction are reviewed and solved analytically in several cases. Direct numerical simulation results for Ginzburg-Landau-Schroedinger and nonlinear wave equations are reported for quantized vortex dynamics and they are compared with those from the reduced dynamics laws.
References:
[1] Y. Zhang, W. Bao and Q. Du, The Dynamics and Interaction of Quantized Vortices in Ginzburg-Landau-Schroedinger equations, SIAM J. Appl. Math., Vol. 67, No. 6, pp. 1740-1775, 2007
[2] A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices
in weakly interacting Bose-Einstein condensates,
Phys. Rev. A, Vol. 76, article 043602, 2007.
[3] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex
dynamics in Ginzburg-Landau-Schrodinger equation,
Eur. J. Appl. Math., Vol. 18, pp. 607-630, 2007.
[4] W. Bao, Q. Du and Y. Zhang, Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numerical computation, SIAM J. Appl. Math., Vol. 66 , No. 3, pp. 758-786, 2006.
[5] W. Bao and Y. Zhang, Dynamics of the ground state and central
vortex states in Bose-Einstein condensation,
Math. Mod. Meth. Appl. Sci., Vol. 15 , No. 12, pp. 1863-1896, 2005.
[6] W. Bao Y. Zhang and R. Zeng, Quantized vortex stability and
interaction in the nonlinear wave equation, preprint.
Interaction-induced
localization of impurities in a Bose-Einstein condensate
Martin Ulrich Bruderer, University of Oxford, UK
Recently, it was pointed out that a neutral impurity atom immersed in a BEC has a localized ground state for sufficiently strong impurity-BEC interactions. Moreover, by varying the impurity-BEC coupling near Feshbach resonances one can control the localization of the impurity and the resulting deformation of the BEC. We show that depending on the strength and, importantly, the sign of the coupling, the impurity state and the deformation of the BEC change considerably. In particular, we find that strongly attractive impurities are highly localized and may lead to the destruction of the BEC, whereas strongly repulsive impurities create a vortex-like BEC state.
Dynamics of vortices and creation of solitons in Bose-Einstein condensates
Matthias Rosenkranz, University of Oxford, UK
We present our recent study of the dynamics of vortices in a Bose-Einstein condensate with weak repulsive interaction [1]. For vortex pairs these weak interactions lead to the breakdown of their separability. We also study other vortex configurations such as the dipole and tripole. The latter configuration shows dynamical creation and annihilation of vortices depending on the initial position of tripole.
We also review some recent results on the creation of solitons in a 1d Bose-Einstein condensate moving through an optical lattice. This system also exhibits a sudden breakdown of the current through the lattice at specific potential heights.
[1] A. Klein, D. Jaksch, Y. Zhang, and W. Bao, Phys. Rev. A 76, 043602
(2007)
A survey on the
development of the Green’s function for the Boltzmann
Equation
Shih-Hsien Yu, National University of Singapore
In this talk we will review the recent development of the Green's function for the Boltzmann equation and its applications to the full nonlinear problem and to the nonlinear stability of the Boltzmann boundary layer.
