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arXiv:cond-mat/0602670 · Journal of Physics B Atomic Molecular and Optical Physics

Beyond mean-field theory for attractive bosons under transverse harmonic confinement

Luca Salasnich

Published Mar 21, 2006• 8 citations
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Abstract

We study a dilute gas of attractive bosons confined in a harmonic cylinder, i.e., under cylindric confinement due to a transverse harmonic potential. We introduce a many-body wavefunction which extends the Bethe ansatz proposed by McGuire (1964 J. Math. Phys. 5 622) by including a variational transverse Gaussian shape. We investigate the ground-state properties of the system comparing them with those of the one-dimensional (1D) attractive Bose gas. We find that the gas becomes ultra 1D as a consequence of the attractive interaction: the transverse width of the Bose gas reduces by increasing the number of particles up to a critical width below which there is the collapse of the cloud. In addition, we derive a simple analytical expression for the symmetry-breaking solitonic density profile of the ground state, which generalizes the one deduced by Calogero and Degasperis (Calogero F and Degasperis A 1975 Phys. Rev. A 11 265). This bright-soliton analytical solution shows near the collapse small deviations with respect to the three-dimensional (3D) mean-field numerical solution. Finally, we show that our variational Gauss–McGuire theory is always more accurate than the McGuire theory. In addition, we prove that, for small numbers of particles, the Gauss–McGuire theory is more reliable than the mean-field theory described by the 3D Gross–Pitaevskii equation.

DOI
10.1088/0953-4075/39/7/016

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We study a dilute gas of attractive bosons confined in a harmonic cylinder, i.e., under cylindric confinement due to a transverse harmonic potential. We introduce a many-body wavefunction which extends the Bethe ansatz proposed by McGuire (1964 J. Math. Phys. 5 622) by including a variational transverse Gaussian shape. We investigate the ground-state properties of the system comparing them with those of the one-dimensional (1D) attractive Bose gas. We find that the gas becomes ultra 1D as a consequence of the attractive interaction: the transverse width of the Bose gas reduces by increasing the number of particles up to a critical width below which there is the collapse of the cloud. In addition, we derive a simple analytical expression for the symmetry-breaking solitonic density profile of the ground state, which generalizes the one deduced by Calogero and Degasperis (Calogero F and Degasperis A 1975 Phys. Rev. A 11 265). This bright-soliton analytical solution shows near the collapse small deviations with respect to the three-dimensional (3D) mean-field numerical solution. Finally, we show that our variational Gauss–McGuire theory is always more accurate than the McGuire theory. In addition, we prove that, for small numbers of particles, the Gauss–McGuire theory is more reliable than the mean-field theory described by the 3D Gross–Pitaevskii equation.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopic
We study a quasi-one-dimensional attractive Bose gas confined in an optical lattice with a superimposed harmonic potential by analyzing the one-dimensional Bose-Hubbard Hamiltonian of the system. Starting from the three-dimensional many-body quantum Hamiltonian, we derive strong inequalities involving the transverse degrees of freedom under which the one-dimensional Bose-Hubbard Hamiltonian can be safely used. To have a reliable description of the one-dimensional ground state, which we call a quantum bright soliton, we use the density-matrix-renormalization-group (DMRG) technique. By comparing DMRG results with mean-field (MF) ones, we find that beyond-mean-field effects become relevant by increasing the attraction between bosons or by decreasing the frequency of the harmonic confinement. In particular, we find that, contrary to the MF predictions based on the discrete nonlinear Schr\"odinger equation, average density profiles of quantum bright solitons are not shape-invariant. We also use the time-evolving-block-decimation method to investigate the dynamical properties of bright solitons when the frequency of the harmonic potential is suddenly increased. This quantum quench induces a breathing mode whose period crucially depends on the final strength of the superimposed harmonic confinement.
We study a gas of bosonic dipolar atoms in the presence of a transverse harmonic trapping potential by using an improved variational Bethe ansatz, which includes the transverse width of the atomic cloud as a variational parameter. Our calculations show that the system behavior evolves from quasi-one dimensional to a strictly one-dimensional one by changing the atom–atom interaction, or the axial density, or the frequency of the transverse confinement. Quite remarkably, in the droplet phase induced by the attractive dipolar interaction the system becomes sub-one dimensional when the transverse width is smaller than the characteristic length of the transverse harmonic confinement.
In this thesis we investigate the properties of ultra-cold quantum gases in reduced dimension and the effects of harmonic confinement on soliton-like properties. We study regimes of agreement between mean-field and many-body theories the generation of entanglement between initially independent finite sized atomic systems. Classical solitons are non-dispersing waves which occur in integrable systems, such as atomic Bose-Einstein condensates in one dimension. Bright and dark solitons are possible, which exist as peaks or dips in density. Quantum solitons are the bound-state solutions to a system satisfying quantum integrability, given via the Bethe Ansatz. Such integrability is broken by the introduction of harmonic confinement. We investigate the equivalence of the classical field and many-body solutions in the limit of large numbers of atoms and derive numerical and variational approaches to examine the ground state energy in harmonic confinement and the fidelity between a Hartree-product solution and a quantum soliton solution. Soliton collisions produce no entanglement between either state and result only in an asymptotic position and phase shift, however external potentials break integrability and thus give the possibility of entangling solitons. We investigate the dynamical entanglement generation between two atomic dimers in harmonic confinement via exact diagonalisation in a basis of Harmonic oscillator functions, making use of the separability of the centre-of-mass component of the Hamiltonian. We show repulsive states show complex dynamics, but with an overall tendency towards states of larger invariant correlation entropy, whereas attractive states resist entanglement unless a phase matching condition is satisfied. This phase matching condition could in theory be used to generate states with highly non-Poissonian number superpositions in atomic systems with controlled number.
We investigate the dynamics of Bose-Einstein condensate bright solitons made of alkali-metal atoms with negative scattering length and under harmonic confinement in the transverse direction. Contrary to the one-dimensional (1D) case, the 3D bright soliton exists only below a critical attractive interaction that depends on the extent of confinement. Such a behavior is also found in multisoliton condensates with box boundary conditions. We obtain numerical and analytical estimates of the critical strength beyond which the solitons do not exist. By using an effective 1D nonpolynomial nonlinear Schr\"odinger equation, which accurately takes into account the transverse dynamics of cigarlike condensates, we numerically simulate the dynamics of the ``soliton train'' reported in a recent experiment [Nature (London) 417, 150 (2002)]. Then, analyzing the macroscopic quantum tunneling of the bright soliton on a Gaussian barrier, we find that its interference in the tunneling region is strongly suppressed with respect to nonsolitonic case; moreover, the tunneling through a barrier breaks the shape invariance of the matter wave. Finally, we show that the collapse of the soliton is induced by the scattering on the barrier or by the collision with another matter wave when the density reaches a critical value, for which we derive an accurate analytical formula.