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arXiv:cond-mat/0411322

Stationary states in a pair of tunnel-coupled two-dimensional condensates with the scattering lengths of opposite sign

V. S. Shchesnovich, Solange Bessa Cavalcanti

Published Nov 12, 2004• 1 citation
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Abstract

We study, analytically and numerically, the stationary states in the system of two linearly coupled nonlinear Schr{ö}dinger equations in two spatial dimensions, with the nonlinear interaction coefficients of opposite signs. This system is the two-dimensional analog of the coupled-mode equations for a condensate in the double-well trap [\textit{Physical Review A} \textbf{69}, 033609 (2004)]. In contrast to the one-dimensional case, where the bifurcation from zero leads to stable bright solitons, in two spatial dimensions this bifurcation results in the appearance of unstable soliton solutions (the Townes-type solitons). With the use of a parabolic potential the ground state of the system is stabilized. It corresponds to strongly coupled condensates and is stable with respect to collapse. This is in sharp contrast to the one-dimensional case, where the ground state corresponds to weakly coupled condensates and is unstable. Moreover, the total number of atoms of the stable ground state can be much higher than the collapse threshold for a single two-dimensional condensate with a negative s-wave scattering length.

DOI
10.48550/arxiv.cond-mat/0411322

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We study, analytically and numerically, the stationary states in the system of two linearly coupled nonlinear Schr{ö}dinger equations in two spatial dimensions, with the nonlinear interaction coefficients of opposite signs. This system is the two-dimensional analog of the coupled-mode equations for a condensate in the double-well trap [\textit{Physical Review A} \textbf{69}, 033609 (2004)]. In contrast to the one-dimensional case, where the bifurcation from zero leads to stable bright solitons, in two spatial dimensions this bifurcation results in the appearance of unstable soliton solutions (the Townes-type solitons). With the use of a parabolic potential the ground state of the system is stabilized. It corresponds to strongly coupled condensates and is stable with respect to collapse. This is in sharp contrast to the one-dimensional case, where the ground state corresponds to weakly coupled condensates and is unstable. Moreover, the total number of atoms of the stable ground state can be much higher than the collapse threshold for a single two-dimensional condensate with a negative s-wave scattering length.
We study solitons in the condensate trapped in a double-well potential with far-separated wells, when the sss-wave scattering length has different signs in the two parts of the condensate. By employing the coupled-mode approximation it is shown that there are unusual stable bright solitons in the condensate, with the larger share of atoms being gathered in the repulsive part. Such unusual solitons derive their stability from the quantum tunneling and correspond to the strong coupling between the parts of the condensate. The ground state of the system, however, corresponds to weak coupling between the condensate parts, with the larger share of atoms being gathered in the attractive part of the condensate.
We predict the existence of an energy barrier for collapse in a system of two tunnel-coupled repulsive and attractive quasi-two-dimensional condensates trapped in a double-well potential. The ground state in such a system can have a lower energy than that of the collapsing state. We find such ground states numerically and analytically determine the domain of their existence; they have a much larger share of atoms confined in the repulsive condensate. The energy barrier for collapse is due to the fact that the energy of the system increases if atoms tunnel from the repulsive condensate to the attractive one.
The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r1/r1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses.
Models of two-dimensional (2D) traps, with double-well structure in the third direction, for Bose-Einstein condensates are introduced with attractive or repulsive interactions between atoms. The models are based on systems of linearly coupled 2D Gross-Pitaevskii equations, where the coupling accounts for tunneling between the wells. Each well carries an optical lattice (OL) (stable 2D solitons cannot exist without OLs). The linear coupling splits each finite band gap in the spectrum of the single-component model into two subgaps. The main subject of the work is spontaneous symmetry breaking (SSB) in two-component 2D solitons and localized vortices (SSB was not considered before in 2D settings). Using variational approximation (VA) and numerical methods, we demonstrate that, in a system with attraction or repulsion, SSB occurs in families of symmetric or antisymmetric solitons (or vortices), respectively. The corresponding bifurcation destabilizes the original solution branch and gives rise to a stable branch of asymmetric solitons or vortices. The VA provides for an accurate description of the emerging branch of asymmetric solitons. In the model with attraction, all stable branches eventually terminate due to the onset of collapse. Stable asymmetric solitons in higher finite band gaps and vortices with a multiple topological charge are found too. The models also give rise to first examples of embedded solitons and embedded vortices (the states located inside Bloch bands) in two dimensions. In the linearly coupled system with opposite signs of the nonlinearity in the two cores, two distinct types of stable solitons and vortices are found, dominated by either the self-attractive component or the self-repulsive one. In the system with a mismatch between the two OLs, a pseudobifurcation is found: when the mismatch attains its largest value $(\ensuremath{\pi})$, the bifurcation does not happen, as branches of different solutions asymptotically approach each other, but fail to merge.
We analyse systematically, from the viewpoint of the nonlinear physics of solitary waves, the effect of the spatial dimension (D = 1,2,3) on the structure and stability of the Bose-Einstein condensates (BECs) trapped in an external anisotropic parabolic potential. While for the positive scattering length the stationary ground-state solutions of the Gross-Pitaevskii equation are shown to be always stable independently of the spatial dimension, for the negative scattering length the ground-state condensate is stable only in the 1D and 2D cases, whereas in the 3D case it becomes unstable. A direct link between nonlinear modes of BECs and (bright and dark) solitary waves of the nonlinear Schrodinger equation is demonstrated for all the dimensions.
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions of the CNLS with a periodic potential are presented and their stability studied by direct numerical simulations. Some of these solutions allow reduction to Manakov system. From a physical point of view these solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against modulations of long wavelength. The solutions which are modulationally unstable are shown to lead to the formation of localized ground states of the coupled BEC system.