1,089 research outputs found
Anisotropic Bose-Einstein condensates and completely integrable dynamical systems
A Gaussian ansatz for the wave function of two-dimensional harmonically
trapped anisotropic Bose-Einstein condensates is shown to lead, via a
variational procedure, to a coupled system of two second-order, nonlinear
ordinary differential equations. This dynamical system is shown to be in the
general class of Ermakov systems. Complete integrability of the resulting
Ermakov system is proven. Using the exact solution, collapse of the condensate
is analyzed in detail. Time-dependence of the trapping potential is allowed
Coupled-mode theory for Bose-Einstein condensates
We apply the concepts of nonlinear guided-wave optics to a Bose-Einstein
condensate (BEC) trapped in an external potential. As an example, we consider a
parabolic double-well potential and derive coupled-mode equations for the
complex amplitudes of the BEC macroscopic collective modes. Our equations
describe different regimes of the condensate dynamics, including the nonlinear
Josephson effect for any separation between the wells. We demonstrate
macroscopic self-trapping for both repulsive and attractive interactions, and
confirm our results by numerical simulations.Comment: 4 pages, 5 figures; typos removed, figures amended; submitted to PR
Vortex states in binary mixture of Bose-Einstein condensates
The vortex configurations in the Bose-Einstein condensate of the mixture of
two different spin states |F=1,m_f=-1> and |2,1> of ^{87}Rb atoms corresponding
to the recent experiments by Matthews et. al. (Phys. Rev. Lett. 83, 2498
(1999)) are considered in the framework of the Thomas-Fermi approximation as
functions of N_2/N_1, where N_1 is the number of atoms in the state |1,-1> and
N_2 - in the state |2,1>. It is shown that for nonrotating condensates the
configuration with the |1,-1> fluid forming the shell about the |2,1> fluid
(configuration "a") has lower energy than the opposite configuration
(configuration "b") for all values of N_2/N_1. When the |1,-1> fluid has net
angular momentum and forms an equatorial ring around the resting central
condensate |2,1>, the total energy of the system is higher than the ground
energy, but the configuration "a" has lower energy than the configuration "b"
for all N_2/N_1. On the other hand, when the |2> fluid has the net angular
momentum, for the lowest value of the angular momentum \hbar l (l=1) there is
the range of the ratio N_2/N_1 where the configuration "b" has lower energy
than the configuration "a". For higher values of the angular momentum the
configuration "b" is stable for all values of N_2/N_1.Comment: minor changes, references adde
Dynamics of a classical gas including dissipative and mean field effects
By means of a scaling ansatz, we investigate an approximated solution of the
Boltzmann-Vlasov equation for a classical gas. Within this framework, we derive
the frequencies and the damping of the collective oscillations of a
harmonically trapped gas and we investigate its expansion after release of the
trap. The method is well suited to studying the collisional effects taking
place in the system and in particular to discussing the crossover between the
hydrodynamic and the collisionless regimes. An explicit link between the
relaxation times relevant for the damping of the collective oscillations and
for the expansion is established.Comment: 4 pages, 1 figur
Solitons in nonlocal nonlinear media: exact results
We investigate the propagation of one-dimensional bright and dark spatial
solitons in a nonlocal Kerr-like media, in which the nonlocality is of general
form. We find an exact analytical solution to the nonlinear propagation
equation in the case of weak nonlocality. We study the properties of these
solitons and show their stability.Comment: 9 figures, submitted to Phys. Rev.
Modulational instability in nonlocal nonlinear Kerr media
We study modulational instability (MI) of plane waves in nonlocal nonlinear
Kerr media. For a focusing nonlinearity we show that, although the nonlocality
tends to suppress MI, it can never remove it completely, irrespectively of the
particular profile of the nonlocal response function. For a defocusing
nonlinearity the stability properties depend sensitively on the response
function profile: for a smooth profile (e.g., a Gaussian) plane waves are
always stable, but MI may occur for a rectangular response. We also find that
the reduced model for a weak nonlocality predicts MI in defocusing media for
arbitrary response profiles, as long as the intensity exceeds a certain
critical value. However, it appears that this regime of MI is beyond the
validity of the reduced model, if it is to represent the weakly nonlocal limit
of a general nonlocal nonlinearity, as in optics and the theory of
Bose-Einstein condensates.Comment: 8 pages, submitted to Phys. Rev.
Collective excitations of a two-dimensional interacting Bose gas in anti-trap and linear external potentials
We present a method of finding approximate analytical solutions for the
spectra and eigenvectors of collective modes in a two-dimensional system of
interacting bosons subjected to a linear external potential or the potential of
a special form , where is the chemical
potential. The eigenvalue problem is solved analytically for an artificial
model allowing the unbounded density of the particles. The spectra of
collective modes are calculated numerically for the stripe, the rare density
valley and the edge geometry and compared with the analytical results. It is
shown that the energies of the modes localized at the rare density region and
at the edge are well approximated by the analytical expressions. We discuss
Bose-Einstein condensation (BEC) in the systems under investigations at and find that in case of a finite number of the particles the regime of BEC
can be realized, whereas the condensate disappears in the thermodynamic limit.Comment: 10 pages, 2 figures include
Stability of trapped Bose-Einstein condensates
In three-dimensional trapped Bose-Einstein condensate (BEC), described by the
time-dependent Gross-Pitaevskii-Ginzburg equation, we study the effect of
initial conditions on stability using a Gaussian variational approach and exact
numerical simulations. We also discuss the validity of the criterion for
stability suggested by Vakhitov and Kolokolov. The maximum initial chirp
(initial focusing defocusing of cloud) that can lead a stable condensate to
collapse even before the number of atoms reaches its critical limit is obtained
for several specific cases. When we consider two- and three-body nonlinear
terms, with negative cubic and positive quintic terms, we have the conditions
for the existence of two phases in the condensate. In this case, the magnitude
of the oscillations between the two phases are studied considering sufficient
large initial chirps. The occurrence of collapse in a BEC with repulsive
two-body interaction is also shown to be possible.Comment: 15 pages, 11 figure
Mean field effects in a trapped classical gas
In this article, we investigate mean field effects for a bosonic gas
harmonically trapped above the transition temperature in the collisionless
regime. We point out that those effects can play also a role in low dimensional
system. Our treatment relies on the Boltzmann equation with the inclusion of
the mean field term.
The equilibrium state is first discussed. The dispersion relation for
collective oscillations (monopole, quadrupole, dipole modes) is then derived.
In particular, our treatment gives the frequency of the monopole mode in an
isotropic and harmonic trap in the presence of mean field in all dimensions.Comment: 4 pages, no figure submitted to Phys. Rev.
Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation
On the basis of recent investigations, a newly developed analytical procedure
is used for constructing a wide class of localized solutions of the controlled
three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the
dynamics of Bose-Einstein condensates (BECs). The controlled 3D GPE is
decomposed into a two-dimensional (2D) linear Schr\"{o}dinger equation and a
one-dimensional (1D) nonlinear Schr\"{o}dinger equation, constrained by a
variational condition for the controlling potential. Then, the above class of
localized solutions are constructed as the product of the solutions of the
transverse and longitudinal equations. On the basis of these exact 3D
analytical solutions, a stability analysis is carried out, focusing our
attention on the physical conditions for having collapsing or non-collapsing
solutions.Comment: 21 pages, 14 figure
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