We consider the threedimensional timedependent grosspitaevskii equation arising in the description of rotating boseeinstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. Boseeinstein condensation and boseeinstein condensation in dilute gases both cover the basic theory at the level of the grosspitaevskii and bogoliubovde gennes equations. Wave packet dynamics for grosspitaevskii equation in one. Mar 18, 20 in particular, we give a rigorous proof of the expansion of the kinetic energy calculated by dalfovo, pitaevskii, and stringari order parameter at the boundary of a trapped bose gas, phys.
A quantum langevin equation for the amplitudes of the collective modes in boseeinstein condensate is derived. In particular, we give a rigorous proof of the expansion of the kinetic energy calculated by dalfovo, pitaevskii, and stringari order parameter at the boundary of a trapped bose gas, phys. To obtain the states, we have made the appropriate transformation to reduce the threedimensional gross. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.
In this description, the quantum kinetic master equation takes the form of a gross pitaevskii equation with noise and damping added according to a well defined prescription an equation we call the stochastic gross pitaevskii equation. Torus quantum vortex knots in the grosspitaevskii model for boseeinstein condensates d. Based on the principle of hamilton dynamics and the principle of lagrangian dynamics, a general pattern formation equation for the spinor boseeinstein condensates is established. Torus quantum vortex knots in the grosspitaevskii model. Mar 14, 2011 the purpose of this paper is to investigate the cauchy problem for the grosspitaevskii infinite linear hierarchy of equations on rn, n. Firstly, we study how does the changes in the trap frequency influence the evolution of the vortices.
From the timedependent grosspitaevskii equation we can look at the dynamics of the boseeinstein condensate. We can see that the vortex patterns of both components are not destroyed, instead the vortex expanded with respect to time. It typically takes 710 iterations only to converge. Expansion of the energy of the ground state of the. Pitaevskii, sandro stringari, alessio recati, arxiv. Barenghi4 1 school of mathematics, university of east anglia, norwich research park, norwich, nr4 7rj, united kingdom 2 dipartimento di fisica, universit a degli studi di torino, via pietro giuria 1, 10125 torino, italy 3 infn, sezione di torino, via pietro giuria 1, 10125. Using the linearized version of the time dependent gross pitaevskii equation we calculate the dynamic response of a boseeinstein condensed gas to periodic density and particle perturbations. From the asymptotic expansion of the ground state of the gross pitaevskii equation in the thomasfermi limit given by gallo and pelinovsky in a previous work, we infer an asymptotic expansion of the kinetic, potential and total energy of the ground state. In the standard literature the gpe is usually obtained in the framework of the second quantization formalism, which in many cases goes beyond the material covered in many advanced undergraduate courses. Torus quantum vortex knots in the grosspitaevskii model for. This cited by count includes citations to the following articles in scholar.
An amended proof of pitaevskiistringaris theorem is given in mathematically coherent manner without resorting to the bogoliubov approximation. Wouters 2005, formation of molecules near a feshbach resonance in a 1d optical lattice. Nonlinear phenomena on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Stringari this book is an introductory text to the physics of boseeinstein condensation. Bifurcation solutions of grosspitaevskii equations for spin. The methods that we have used here are taken by pursuing the recent work of kivshar et. Boseeinstein condensation by pitaevskii and coauthor sandro stringari, who is another major contributor to the recent theoretical literature, reflects the brief history just sketched. Bifurcation solutions of grosspitaevskii equations for. Superfluidity of polaritons in semiconductor microcavities.
But pitaevskii and stringaris book places less emphasis on what one might call the atomic physics aspects of dilute alkali gases. Jan 23, 2019 using the linearized version of the time dependent grosspitaevskii equation, we calculate the dynamic response of a boseeinstein condensed gas to periodic density and particle perturbations. Now we complete our investigation on the validity of using the tssp for solving the gpe by comparing the results obtained in section 4. Oct 23, 2000 read stationary solutions of the grosspitaevskii equation with linear counterpart, physics letters a on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. We need to increase the value of c to at least c30 to make the procedure work. Boseeinstein condensation and superfluidity in trapped. Gap solitons for the repulsive grosspitaevskii equation with. After a few years spent at the israel institute of technology in haifa he eventually became professor of trento university in 1998. He has developed in a systematic way the sum rule approach to. Gross and lev petrovich pitaevskii describes the ground state of a quantum system of identical bosons using the hartreefock approximation and the pseudopotential interaction model in the hartreefock approximation the total wavefunction of the system of bosons is taken as a product of singleparticle functions. Bifurcation analysis of stationary solutions of two. First, cooling of neutral atoms into their motional ground state, thus.
The grosspitaevskii equation, also called the nonlinear schr. Amended proof of pitaevskiistringaris theorem request pdf. The grosspitaevskii equation and boseeinstein condensates 249 of the system is the creation of vortices, which can be seen as small twisters inside the liquid that behave quite differently. A timesplitting spectral method for coupled grosspitaevskii. Avramenko, physica d, 254, 29 20 that under certain conditions all the stationary modes for the 1d gpe can be coded by biinfinite. Gap solitons for the repulsive grosspitaevskii equation. This volume introduces the basic concepts of boseeinstein condensation and superfluidity. The paper is devoted to nonlinear localized modes gap solitons for the spatially onedimensional grosspitaevskii equation 1d gpe with a periodic potential and repulsive interparticle interactions. The same analysis is also carried out at even higher angular velocities, where the system enters the giant vortex regime. The resulting effective equations in one or two spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. Buy sandro stringari ebooks to read online or download in pdf or epub on your pc, tablet or mobile device. Among them, nonlinear wave dynamical features are of particular interest in the form of bright and dark solitons, vortices, and vortex lines and vortex rings. Wolfgang pauli institute 2006 thematic program januarydecember, 2006 vienna, austria contemporary mathematics on free shipping on qualified orders.
A 54, 424217 1996 in the case where the space dimension is 3. Lev pitaevskii and sandro stringari, oxford university press, oxford, 2003. Boseeinstein condensates confined in traps exhibit unique features which have been the object of extensive experimental and theoretical studies in th. The grosspitaevskii equation and boseeinstein condensates. This phenomenon, first predicted by einstein in 1925, has been realized experimentally in 1995 in a remarkable series of experiments whose. Download the bible, the holy quran, the mahabharata and thousands of free pdf ebooks on buddhism, meditation etc. A natural generalization of the heisenberg uncertainty principle inequality holding for non hermitian operators is presented and applied to the fractional quantum hall effect fqhe.
Boseeinstein condensation represents a new state of matter and is one of the cornerstones of quantum physics, resulting in the 2001 nobel prize. Since the grosspitaevskii equation is a nonlinear partial differential equation, exact solutions are hard to come by. Sandro stringari university of trento 1 introduction to boseeinstein condensation 3. This book is an introductory text to the physics of boseeinstein condensation. Note that larger values of k require larger values of the constant c for the algorithm to converge. Torus quantum vortex knots in the gross pitaevskii model for bose. Questo conteggio citato da include citazioni ai seguenti articoli in scholar. Gross pitaevskii equation for the order parameter and beyond cnrinfm.
An amended proof of pitaevskii stringari s theorem is given in mathematically coherent manner without resorting to the bogoliubov approximation. Read stationary solutions of the grosspitaevskii equation with linear counterpart, physics letters a on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Download fulltext pdf grosspitaevskii dynamics of boseeinstein condensates and superfluid turbulence article pdf available in fluid dynamics research 335. Theory of boseeinstein condensation in trapped gases. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis.
Gross pitaevskii gp equation for order parameter plays role analogous to maxwell equations in classical electrodynamics. The onedimensional grosspitaevskii equation and its some. The quest for superfluidity in fermi gases science. Download the complete works of fjodor dostojevskij free pdfs. Read formal analytical solutions for the grosspitaevskii equation, physica d. Our approach points out the crucial role played by. Quantum fluctuations and grosspitaevskii theory springerlink. We suggest a simple gaussian lagrangian variational scheme for the reduced timedependent quasioneand quasitwodimensional grosspitaevskii gp equations of a dipolar boseeinstein condensate bec in cigar and disk configurations, respectively. Providing a useful introduction to one of the most exciting field of physics today, this text will be of interest. The ones marked may be different from the article in the profile. We have derived some excitation states of the onedimensional grosspitaevskii equation coupled by the gravitational potential.
The gross pitaevskii model of spinor boseeinstein condensates is studied. This inequality was used in a previous paper to prove the absence of long range order in the ground state of several 1d systems with continuous group symmetries. Mar 14, 2002 the early experiments on boseeinstein condensation in dilute atomic gases accomplished three longstanding goals. Compressibility, bloch bands and elementary excitations. A striking consequence of the appearance of macroscopic coherence in.
It is predicted analytically and verified numerically that for a free particle with attractive interaction, the dynamics of the width is governed by an effective potential which is sensitive to initial conditions. The zero temperature limit of the fluctuationdissipation theorem is used to evaluate the corresponding quantum fluctuations induced by the. Using the linearized version of the time dependent grosspitaevskii equation we calculate the dynamic response of a boseeinstein condensed gas to periodic density and particle perturbations. On the cauchy problem for grosspitaevskii hierarchies. Discover book depositorys huge selection of l p pitaevskii books online. We provide an overview of the recent theoretical and experimental advances in the study of second sound in ultracold atomic gases. Formal analytical solutions for the grosspitaevskii. Since no analytic solutions of the grosspitaevskii equation are known to us, we use the dual or dirac stepping ds method in two dimension as well as the thomasfermi tf approximation for numerical comparisons. From the asymptotic expansion of the ground state of the grosspitaevskii equation in the thomasfermi limit given by gallo and pelinovsky in a previous work, we infer an asymptotic expansion of the kinetic, potential and total energy of the ground state. Barenghi4 1 school of mathematics, university of east anglia, norwich research park, norwich, nr4.
This phenomenon, first predicted by einstein in 1925 has been realized experimentally in 1995 in a remarkable series of experiments whose importance has been recognized by the award of the 2001 nobel prize in physics. Starting from the landau two fluid hydrodynamic equations we develop the theory of first and second sound in various configurations characterized by different geometries and quantum statistics. The variational approximation for stationary states and breathing oscillation dynamics in reduced dimensions agrees well with the. Apr 04, 2016 download free pdf ebooks download pdfs. In what follows, students will learn how to use the ndsolve command of mathematica. The remainder is devoted specifically to the dilute alkali gases. Dimension reduction for rotating boseeinstein condensates. The stochastic grosspitaevskii equation iopscience. Buy stationary and time dependent grosspitaevskii equations. Expansion of the energy of the ground state of the gross. Quelli contrassegnati con possono essere diversi dallarticolo nel profilo.
Langevin equation of collective modes of boseeinstein. Meanfield theory provides a framework to understand the main features of the condensation and the role of interactions between particles. The book is an introductory text to the physics of boseeinstein condensation. This approach is based on the orthodox quantum field theory which rigorously maintains the canonical commutation relations. Stringari, boseeinstein condensation oxford university press, oxford, 2003. The propagation of an initially gaussian wave packet of width.
Stationary solutions of the grosspitaevskii equation with. The collective modes are coupled to a thermal reservoir of quasiparticles, whose elimination leads to the quantum langevin equation. Boseeinstein condensation and superfluidity oxford scholarship. A 100, 023607 2019 supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas. Buy lev pitaevskii ebooks to read online or download in pdf or epub on your pc, tablet or mobile device.
Lev pitaevskii and sandro stringari, oxford university press, oxford, 2003 find, read and. We prove local existence and uniqueness of solutions in certain sobolevtype spaces h of sequences of marginal density operators with. The gross pitaevskii equation and boseeinstein condensates 249 of the system is the creation of vortices, which can be seen as small twisters inside the liquid that behave quite differently. Read a timesplitting spectral method for coupled grosspitaevskii equations with applications to rotating boseeinstein condensates, journal of computational and applied mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Furthermore, three conservative quantities of spinor bec, that is, the energy, total particle number, and magnetization intensity, are also proved. A fast algorithm for the solution of the timeindependent. Sandro stringari is an italian theoretical physicist, who has contributed to the theory of quantum manybody physics, including atomic nuclei, quantum liquids and ultracold atomic bose and fermi gases. This phenomenon, first predicted by einstein in 1925, has been realized experimentally in 1995 in a remarkable series of experiments whose importance has been recognized by the award of the 2001 nobel prize in physics.
The grosspitaevskii equation gpe, named after eugene p. Buy stationary and time dependent gross pitaevskii equations. In their perspective, pitaevskii and stringari discuss recent studies of the. Lev pitaevskii started collaborating with the university of trento at the end of the 1980s through a series of long term visits. From this, a very simplified description we call the phenomenological growth equation can be derived. Selftrapping and josephson tunneling solutions to the. Sep 20, 2009 similar to atoms in cold gases, excitonpolaritons in semiconductor microcavities can undergo boseeinstein condensation. Stationary and time dependent grosspitaevskii equations. Landau damping in weakly interacting bose gases is investigated by means of perturbation theory.
The main aim of this paper is to study the bifurcation solutions associated with the spinor boseeinstein condensates. The study of boseeinstein condensates becs has offered a versatile playground for the examination of a diverse host of mesoscopic quantum phenomena for more than two decades. Uncertainty principle and offdiagonal long range order in. Torus quantum vortex knots in the gross pitaevskii model for boseeinstein condensates d. Aug 01, 2007 read a timesplitting spectral method for coupled grosspitaevskii equations with applications to rotating boseeinstein condensates, journal of computational and applied mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. This phenomenon, first predicted by einstein in 1925 has been realized experimentally in 1995 in a remarkable series of experiments whose importance has been recognized. Clarendon press, apr 3, 2003 literary criticism 382 pages. Pdf the grosspitaevskii equation and boseeinstein condensates. It is used to find the collective modes of a trapped gas. We propose a timesplitting spectral method for the coupled grosspitaevskii equations, which describe the dynamics of rotating twocomponent boseeinstein condensates at a very low temperature. Increasing c, however, affects the rate of convergence of. Using the abstract results obtained for infinitedimensional hamilton system, we establish the mathematical theory for the model of spinor bec. Moreover, we calculate one more term in this expansion, and we generalize.
The purpose of this paper is to investigate the cauchy problem for the grosspitaevskii infinite linear hierarchy of equations on rn, n. Submitted on 29 may 2018 v1, last revised 17 sep 2018 this version, v2. The zero temperature limit of the fluctuationdissipation theorem is used to evaluate the corresponding quantum fluctuations induced by the elementary excitations in the ground state. Nlsgp governs nearlymonochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear kerr refractive indices and zero absorption, as well as the behavior of boseeinstein condensates. In particular, we give a clear discussion of all cases. Using the linearized version of the time dependent grosspitaevskii equation, we calculate the dynamic response of a boseeinstein condensed gas to periodic density and particle perturbations.
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