Vortex formation in a slowly rotating Bose-Einstein condensate confined in a harmonic-plus-Gaussian laser trap

Motivated by the recent experiment at ENS [V. Bretin, S. Stock, Y. Seurin and, J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004)], we study a rotating (non-)interacting atomic Bose-Einstein condensate confined in a harmonic-plus-Gaussian laser trap potential. By adjusting the amplitude of the Gaussian laser potential, one can make quadratic-plus-quartic potential, purely quartic potential, and quartic-minus-quadratic potential. We show that an interacting Bose-Einstein condensate confined in a harmonic-plus-Gaussian laser trap breaks the rotational symmetry of the Hamiltonian when rotational frequency is greater than one-half of the lowest energy surface mode frequency. We also show that by increasing the amplitude of the Gaussian laser trap, a vortex appears in a slowly rotating Bose-Einstein condensate. Moreover, one can also create a vortex in a slowly rotating non-interacting Bose-Einstein condensate confined in harmonic-plus-Gaussian laser potential.Received: 24 June 2004, Published online: 24 August 2004PACS: 03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices and topological excitations - 05.30.Jp Boson systems     

Phases of atom-molecule vortex matter

We study ground state vortex configurations in a rotating atom-molecule Bose-Einstein condensate. It is found that the coherent coupling between the atomic and molecular condensates can render a pairing of atomic and molecular vortices into a composite structure that resembles a carbon dioxide molecule. Structural phase transitions of vortex lattices are also explored through different physical parameters including the rotational frequency of the system.     

Photonic band gap via quantum coherence in vortex lattices of Bose-Einstein condensates

We investigate the optical response of an atomic Bose-Einstein condensate with a vortex lattice. We find that it is possible for the vortex lattice to act as a photonic crystal and create photonic band gaps, by enhancing the refractive index of the condensate via a quantum coherent scheme. If high enough index contrast between the vortex core and the atomic sample is achieved, a photonic band gap arises depending on the healing length and the lattice spacing. A wide range of experimentally accessible parameters are examined and band gaps in the visible region of the electromagnetic spectrum are found. We also show how directional band gaps can be used to directly measure the rotation frequency of the condensate.     

Bose-Einstein condensates with a bent vortex in rotating traps

We consider a 3D dilute Bose-Einstein condensate at thermal equilibrium in a rotating harmonic trap. The condensate wavefunction is a local minimum of the Gross-Pitaevskii energy functional and we determine it numerically with the very efficient conjugate gradient method. For single vortex configurations in a cigar-shaped harmonic trap we find that the vortex line is bent, in agreement with the numerical prediction of Garcia-Ripoll and Perez-Garcia [Phys. Rev. A 63, 041603 (2001)]. We derive a simple energy functional for the vortex line in a cigar-shaped condensate which allows to understand physically why the vortex line bends and to predict analytically the minimal rotation frequency required to stabilize the bent vortex line. This analytical prediction is in excellent agreement with the numerical results. It also allows to find in a simple way a saddle point of the energy, where the vortex line is in a stationary configuration in the rotating frame but not a local minimum of energy. Finally we investigate numerically the effect of thermal fluctuations on the vortex line for a condensate with a straight vortex: we can predict what happens in a single realization of the experiment by a Monte Carlo sampling of an atomic field quasi-distribution function of the density operator of the gas at thermal equilibrium in the Bogoliubov approximation. Received 28 March 2002 / Received in final form 13 September 2002 Published online 21 January 2003 RID="a" ID="a"e-mail: yvan.castin@lkb.ens.fr     

Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.     

谐振子势与高斯势联合势阱中玻色爱因斯坦凝聚体的巨涡旋态

研究谐振子势与高斯势联合势阱中玻色爱因斯坦凝聚体的基态。发现凝聚体形成巨涡旋时,其涡旋个数等于平均角动量,且凝聚体密度分布和角动量密度分布相同,进而得到凝聚体形成巨涡旋时所处基态是角动量的本征态。发现势阱从各向同性的环形势阱逐渐变为各向异性的环形势阱的过程中,凝聚体的平均角动量与涡旋个数之比先由1平缓下降,然后迅速下降,最后保持在0.5附近。同时给出凝聚体密度分布和角动量分布的特征,并作出相应解释。     

Transfer and storage of vortex states in light and matter waves

We theoretically explore the transfer of vortex states between atomic Bose-Einstein condensates and optical pulses using ultraslow and stopped light techniques. We find shining a coupling laser on a rotating two-component ground state condensate with a vortex lattice generates a probe laser field with optical vortices. We also find that optical vortex states can be robustly stored in the atomic superfluids for times, in Rb-87 condensates, limited only by the ground state coherence time.     

Atom optics with rotating Bose-Einstein condensates

The atom optics of Bose-Einstein condensates containing a vortex of circulation one is discussed. We first analyze in detail the reflection of such a condensate falling on an atomic mirror. In a second part, we consider a rotating condensate in the case of attractive interactions. We show that for sufficiently large nonlinearity the rotational symmetry of the rotating condensate is broken. Received 16 September 2002 / Received in final form 17 November 2002 Published online 11 February 2003     

Emergence and decay of turbulence in stirred atomic Bose-Einstein condensates

We show that "weak" elliptical deformation of an atomic Bose-Einstein condensate rotating at close to the quadrupole instability frequency leads to turbulence with a Kolmogorov energy spectrum. The turbulent state is produced by energy transfer to condensate fragments that are ejected by the quadrupole instability. This energy transfer is driven by breaking the twofold rotational symmetry of the condensate. Subsequently, vortex-sound interactions damp the turbulent state leading to the crystallization of a vortex lattice.     

Quantum condensates

We analyze the atomic state obtained by photo-dissociation of a molecular condensate. We show that the dissociation process produces an atomic condensate in a squeezed state in analogy with the down-conversion process leading to squeezed light in quantum optics.     

Vortex rings and solitary waves in trapped Bose–Einstein condensates

We discuss nonlinear excitations in an atomic Bose–Einstein condensate which is trapped in a harmonic potential. We focus on axially symmetric solitary waves propagating along a cylindrical condensate. A quasi one-dimensional dark soliton is the only nonlinear mode for a condensate with weak interactions. For sufficiently strong interactions of experimental interest solitary waves are hybrids of one-dimensional dark solitons and three-dimensional vortex rings. The energy-momentum dispersion of these solitary waves exhibits characteristics similar to a mode proposed sometime ago by Lieb in a strictly 1D model, as well as some rotonlike features. We subsequently discuss interactions between solitary waves. Head-on collisions between dark solitons are elastic. Slow vortex rings collide elastically but faster ones form intermediate structures during collisions before they lose energy to the background fluid. Solitary waves and their interactions have been observed in experiments. However, some of their intriguing features still remain to be experimentally identified.     

Vortex pump for dilute bose-einstein condensates

The formation of vortices by topological phase engineering has been realized experimentally to create the first two- and four-quantum vortices in dilute atomic Bose-Einstein condensates. We consider a similar system, but in addition to the Ioffe-Pritchard magnetic trap we employ an additional hexapole field. By controlling cyclically the strengths of these magnetic fields, we show that a fixed amount of vorticity can be added to the condensate in each cycle. In an adiabatic operation of this vortex pump, the appearance of vortices into the condensate is interpreted as the accumulation of a local Berry phase. Our design can be used as an experimentally realizable vortex source for possible vortex-based applications of dilute Bose-Einstein condensates.     

Ultracold superstrings in atomic boson-fermion mixtures

We propose a setup with ultracold atomic gases that can be used to make a nonrelativistic superstring in four spacetime dimensions. In particular, we consider for the creation of the superstring a fermionic atomic gas that is trapped in the core of a vortex in a Bose-Einstein condensate. We explain the required tuning of experimental parameters to achieve supersymmetry between the fermionic atoms and the bosonic modes describing the oscillations in the vortex position. Furthermore, we discuss the experimental consequences of supersymmetry.     

Tunneling Dynamics of Two-Species Molecular Bose-Einstein Condensates

We study tunneling dynamics of atomic group in two-species molecular Bose-Einstein condensates. It is shown that the tunneling of the atom group depends on not only the tunneling coupling constant between the atomic pair molecular condensate and the three-atomic group molecular condensate, but also the inter-molecular nonlinear interactions and the initial number of atoms in these condensates. It is discovered that besides oscillating tunneling current between the atomic pair molecular condensate and the three-atomic group molecular condensate, the nonlinear atomic group tunneling dynamics sustains a self-maintained population imbalance: a macroscopic quantum self-trapping effect.     

A Rotating Bose-Einstein Condensation in a Toroidal Trap

We have studied the ground state configurations of a rotating Bose-Einstein condensation in a toroidal trap as the radius of the central Gaussian potential expands adiabatically. Firstly, we observe that the vortices are devoured successively into the central hole of the condensate to form a giant vortex as the radius of the trap expands. When all the pre-existing vortices are absorbed, the angular momentum of the system still increase as the radius of the
gaussian potential enlarges. When increasing the interaction strength, we find that more singly quantized vortices are squeezed into the condensate, but the giant vortex does not change.     

Spatially periodic structures of an atomic Bose-Einstein condensate

The conditions providing the formation of periodic vortex lattices of an interference nature in an atomic Bose-Einstein condensate (i.e., in the absence of rotation of the condensate) are determined. Spatially periodic exact solutions of the nonlocal nonlinear Schrödinger equation (the generalized Gross-Pitaevskii equation) that describes the Bose-Einstein condensate of a dilute gas of alkali metal atoms with due regard for the nonlocality of interatomic interactions are obtained in the form of a set of two or three plane waves. It is shown that periodic vortex lattices can be produced in interference experiments with a Bose-Einstein condensate of a dilute gas of alkali metal atoms.     

Creating vortex rings and three-dimensional skyrmions in Bose-Eeinstein condensates

We propose a method of generating a vortex ring in a Bose-Einstein condensate by means of electromagnetically induced atomic transitions. Some remnant population of atoms in a second internal state remains within the toroidal trap formed by the mean-field repulsion of the vortex ring. This population can be removed, or it can be made to flow around the torus (i.e., within the vortex ring). If this flow has a unit topological winding number, the entire structure formed by the two condensates is an example of a three-dimensional Skyrmion texture.     

Quantum phase transition in an atomic bose gas with a feshbach resonance

We show that in an atomic Bose gas near a Feshbach resonance a quantum phase transition occurs between a phase with only a molecular Bose-Einstein condensate and a phase with both an atomic and a molecular Bose-Einstein condensate. We show that the transition is characterized by an Ising order parameter. We also determine the phase diagram of the gas as a function of magnetic field and temperature: the quantum critical point extends into a line of finite temperature Ising transitions.     

Structure and stability of vortices in dilute Bose-Einstein condensates at ultralow temperatures

We compute the structure of a quantized vortex line in a harmonically trapped dilute atomic Bose-Einstein condensate using the Popov version of the Hartree-Fock-Bogoliubov mean-field theory. The vortex is shown to be (meta)stable in a nonrotating trap even in the zero-temperature limit, thus confirming that weak particle interactions induce for the condensed gas a fundamental property characterizing "classical" superfluids. We present the structure of the vortex at ultralow temperatures and discuss the crucial effect of the thermal gas component to its energetic stability.     

Quantized rotation of atoms from photons with orbital angular momentum

We demonstrate the coherent transfer of the orbital angular momentum of a photon to an atom in quantized units of variant Planck's over 2pi, using a 2-photon stimulated Raman process with Laguerre-Gaussian beams to generate an atomic vortex state in a Bose-Einstein condensate of sodium atoms. We show that the process is coherent by creating superpositions of different vortex states, where the relative phase between the states is determined by the relative phases of the optical fields. Furthermore, we create vortices of charge 2 by transferring to each atom the orbital angular momentum of two photons.