Bose-Einstein condensates with attractive interactions on a ring

Considering an effectively attractive quasi-one-dimensional Bose-Einstein condensate of atoms confined in a toroidal trap, we find that the system undergoes a phase transition from a uniform to a localized state, as the magnitude of the coupling constant increases. Both the mean-field approximation, as well as a diagonalization scheme are used to attack the problem.Comment: 4 pages, 4 ps figures, RevTex, typographic errors correcte

Instabilities in the two-dimensional cubic nonlinear Schrodinger equation

The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. In this paper, we establish that every one-dimensional traveling wave solution of NLS with trivial phase is unstable with respect to some infinitesimal perturbation with two-dimensional structure. If the coefficients of the linear dispersion terms have the same sign then the only unstable perturbations have transverse wavelength longer than a well-defined cut-off. If the coefficients of the linear dispersion terms have opposite signs, then there is no such cut-off and as the wavelength decreases, the maximum growth rate approaches a well-defined limit.Comment: 4 pages, 4 figure

Stability of Repulsive Bose-Einstein Condensates in a Periodic Potential

The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schr\"odinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.Comment: 12 pages, 17 figure

Stability of Attractive Bose-Einstein Condensates in a Periodic Potential

Using a standing light wave trap, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.Comment: 12 pages, 18 figure

Nanoengineering Carbon Allotropes from Graphene

Monolithic structures can be built into graphene by the addition and subsequent re-arrangement of carbon atoms. To this end, ad-dimers of carbon are a particularly attractive building block because a number of emerging technologies offer the promise of precisely placing them on carbon surfaces. In concert with the more common Stone-Wales defect, repeating patterns can be introduced to create as yet unrealized materials. The idea of building such allotropes out of defects is new, and we demonstrate the technique by constructing two-dimensional carbon allotropes known as haeckelite. We then extend the idea to create a new class of membranic carbon allotropes that we call \emph{dimerite}, composed exclusively of ad-dimer defects.Comment: 5 pages, 5 figure

Effectively attractive Bose-Einstein condensates in a rotating toroidal trap

We examine an effectively attractive quasi-one-dimensional Bose-Einstein condensate of atoms confined in a rotating toroidal trap, as the magnitude of the coupling constant and the rotational frequency are varied. Using both a variational mean-field approach, as well as a diagonalization technique, we identify the phase diagram between a uniform and a localized state and we describe the system in the two phases.Comment: 4 pages, 4 ps figures, RevTe

Demkov-Kunike model for cold atom association: weak interaction regime

We study the nonlinear mean-field dynamics of molecule formation at coherent photo- and magneto-association of an atomic Bose-Einstein condensate for the case when the external field configuration is defined by the quasi-linear level crossing Demkov-Kunike model, characterized by a bell-shaped pulse and finite variation of the detuning. We present a general approach to construct an approximation describing the temporal dynamics of the molecule formation in the weak interaction regime and apply the developed method to the nonlinear Demkov-Kunike problem. The presented approximation, written as a scaled solution to the linear problem associated to the nonlinear one we treat, contains fitting parameters which are determined through a variational procedure. Assuming that the parameters involved in the solution of the linear problem are not modified, we suggest an analytical expression for the scaling parameter.Comment: 6 pages, 4 figure

Ultracold Atoms in 1D Optical Lattices: Mean Field, Quantum Field, Computation, and Soliton Formation

In this work, we highlight the correspondence between two descriptions of a system of ultracold bosons in a one-dimensional optical lattice potential: (1) the discrete nonlinear Schr\"{o}dinger equation, a discrete mean-field theory, and (2) the Bose-Hubbard Hamiltonian, a discrete quantum-field theory. The former is recovered from the latter in the limit of a product of local coherent states. Using a truncated form of these mean-field states as initial conditions, we build quantum analogs to the dark soliton solutions of the discrete nonlinear Schr\"{o}dinger equation and investigate their dynamical properties in the Bose-Hubbard Hamiltonian. We also discuss specifics of the numerical methods employed for both our mean-field and quantum calculations, where in the latter case we use the time-evolving block decimation algorithm due to Vidal.Comment: 14 pages, 2 figures; to appear in Journal of Mathematics and Computers in Simulatio

Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the Nonlinear Schr\"odinger equation and tunneling splittings

We present a representative set of analytic stationary state solutions of the Nonlinear Schr\"odinger equation for a symmetric double square well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry preserving even and odd states, nonlinearity introduces quite exotic symmetry breaking solutions - among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry breaking localized solutions to form macroscopic quantum superpositions states and explore a simple model for the exponentially small tunneling splitting.Comment: 11 pages, 11 figures, revised version, typos and references correcte