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 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.
We introduce a one-dimensional spatially inhomogeneous Bose-Hubbard model (BHM) with the strength of the onsite repulsive interactions growing, with the discrete coordinate
zj, as $|{z}_{j}{|}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>0$. Recently, the analysis of the mean-field (MF) counterpart of this system has demonstrated self-trapping of robust unstaggered discrete solitons, under the condition $\ensuremath{\alpha}>1$. By using the numerically implemented method of the density matrix renormalization group, we demonstrate that, in a certain range of the interaction, the BHM also features self-trapping of the ground state into a soliton-like configuration, at $\ensuremath{\alpha}>1$, and remains weakly localized at $\ensuremath{\alpha}<1$. An essential quantum feature found in the BHM is a residual quasi-constant density of the background surrounding the soliton-like peak in the ground state, while in the MF limit the finite-density background is absent. Very strong onsite repulsion eventually destroys soliton-like states, driving the system, at integer densities, into the Mott phase with a spatially uniform density.
Within the framework of a mean-field description, we investigate atomic Bose-Einstein condensates, with attraction between atoms, under the action of a strong transverse confinement and periodic [optical-lattice (OL)] axial potential. Using a combination of the variational approximation, one-dimensional (1D) nonpolynomial Schr\"odinger equation, and direct numerical solutions of the underlying 3D Gross-Pitaevskii equation, we show that the ground state of the condensate is a soliton belonging to the semi-infinite band gap of the periodic potential. The soliton may be confined to a single cell of the lattice or extended to several cells, depending on the effective self-attraction strength
g (which is proportional to the number of atoms bound in the soliton) and depth of the potential,
V0, the increase of
V0 leading to strong compression of the soliton. We demonstrate that the OL is an effective tool to control the soliton's shape. It is found that, due to the 3D character of the underlying setting, the ground-state soliton collapses at a critical value of the strength,
g=gc, which gradually decreases with the increase of
V0; under typical experimental conditions, the corresponding maximum number of
7Li atoms in the soliton,
Nmax, ranges between 8000 and 4000. Examples of stable multipeaked solitons are also found in the first finite band gap of the lattice spectrum. The respective critical value
gc again slowly decreases with the increase of
V0, corresponding to ${N}_{\mathrm{max}}\ensuremath{\simeq}5000$.
Consider the tunneling of a many-body wavefunction through a potential barrier. Specifically we examine quantum many-body tunneling of ultracold bosons in one dimensional optical lattice potentials. Such systems are described by the Bose-Hubbard model, and serve as an ideal testbed for the observation of quantum phenomena on macroscopic length scales. Bright solitons, self-sustaining nonlinear waves, are confined by a thin potential barrier. By decreasing the size of the barrier or by tuning the nonlinearity, we find that the bound states can be transformed into quasi-bound ones as the soliton tunnels collectively out of confinement. The lowly entangled 1D many-body problem can be numerically solved via time-evolving block decimation: a time-adaptive density matrix renormalization group routine. As a sanity check the results can be compared to mean-field theory predictions obtained from numerical analysis of the Nonlinear Schrodinger equation. The goal behind this effort is to explore the previously intractable physics in regimes where the mean-field theory fails, and to provide a stringent test on the validity of quantum mechanics for complex entangled systems.
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.
We present a fully quantum many-body treatment of dark solitons formed by ultracold bosonic atoms in one-dimensional optical lattices. Using time-evolving block decimation to simulate the single-band Bose-Hubbard Hamiltonian, we consider the quantum dynamics of density and phase engineered dark solitons as well as the quantum evolution of mean-field dark solitons injected into the quantum model. The former approach directly models how one may create quantum entangled dark solitons in experiment. While we have already presented results regarding the latter approach elsewhere [R. V. Mishmash and L. D. Carr, Phys. Rev. Lett. 103, 140403 (2009)], we expand upon those results in this work. In both cases, quantum fluctuations cause the dark soliton to fill in and may induce an inelasticity in soliton-soliton collisions. Comparisons are made to the Bogoliubov theory which predicts depletion into an anomalous mode that fills in the soliton. Our many-body treatment allows us to go beyond the Bogoliubov approximation and calculate explicitly the dynamics of the system's natural orbitals.