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arXiv:0801.2999 · Physical Review A

Nonlinear intraband tunneling of a Bose-Einstein condensate in a cubic three-dimensional lattice

V. S. Shchesnovich, V. V. Konotop

Published Jan 23, 2008• 9 citations
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Abstract

The intraband tunneling of a Bose-Einstein condensate between three degenerate high-symmetry $X$ points of the Brillouin zone of a cubic optical lattice is studied in the quantum regime by reduction to a three-mode model. The mean-field approximation of the deduced model is described. Compared to the previously reported two-dimensional (2D) case [V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 75, 063628 (2007)], which is reducible to the two-mode model, in the case under consideration there exist a number of new stable stationary atomic distributions between the $X$ points and a new critical lattice parameter. The quantum collapses and revivals of the atomic population dynamics are absent for the experimentally realizable time span. The 2D stationary configurations, embedded into the 3D lattice, turn out to be always unstable, while the existence of a stable 1D distribution, where all atoms populate only one $X$ state, may serve as a starting point in an experimental study of nonlinear tunneling in a 3D lattice.

DOI
10.1103/physreva.77.013614

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The intraband tunneling of a Bose-Einstein condensate between three degenerate high-symmetry XXX points of the Brillouin zone of a cubic optical lattice is studied in the quantum regime by reduction to a three-mode model. The mean-field approximation of the deduced model is described. Compared to the previously reported two-dimensional (2D) case [V. S. Shchesnovich and V. V. Konotop, Phys. Rev. A 75, 063628 (2007)], which is reducible to the two-mode model, in the case under consideration there exist a number of new stable stationary atomic distributions between the XXX points and a new critical lattice parameter. The quantum collapses and revivals of the atomic population dynamics are absent for the experimentally realizable time span. The 2D stationary configurations, embedded into the 3D lattice, turn out to be always unstable, while the existence of a stable 1D distribution, where all atoms populate only one XXX state, may serve as a starting point in an experimental study of nonlinear tunneling in a 3D lattice.
Quantum theory of the intraband resonant tunneling of a Bose-Einstein condensate loaded in a two-dimensional optical lattice is considered. It is shown that the phenomena of quantum collapse and revival can be observed in the fully quantum problem. The mean-field limit of the theory is analyzed using the WKB approximation for discrete equations, establishing in this way a direct connection between the two approaches conventionally used in very different physical contexts. More specifically we show that there exist two different regimes of tunneling and study dependence of quantum collapse and revival on the number of condensed atoms.
We present a thorough analysis of the nonlinear tunneling of Bose-Einstein condensates in static and accelerating two-dimensional lattices within the framework of the mean-field approximation. We deal with nonseparable lattices, considering different initial atomic distributions in highly symmetric states. For an analytical description of the condensate before instabilities develop, we derive several few-mode models, analyzing essentially both nonlinear and quasilinear regimes of tunneling. By direct numerical simulations, we show that two-mode models provide an accurate description of tunneling when either initially two states are populated or tunneling occurs between two stable states. Otherwise, a two-mode model may give only useful qualitative hints for understanding tunneling, but does not reproduce many features of the phenomenon. This reflects the crucial role of instabilities developed due to two-body interactions resulting in a non-negligible population of the higher bands. This effect becomes even more pronounced in the case of accelerating lattices. In the latter case we show that the direction of the acceleration is a relevant physical parameter which affects the tunneling by changing the atomic rates at different symmetric states and by changing the numbers of bands involved in the atomic transfer.
We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.
The dynamics of localized excitations in an array of Bose-Einstein condensates (BECs) is investigated in the framework of the nonlinear lattice theory. The existence of temporarily stable ground states displaying an atomic population distribution localized on very few lattice sites (intrinsic localized modes), as well as atomic population distributions involving many lattice sites (envelope solitons), is studied both numerically and analytically. The origin and properties of these modes are shown to be inherently connected with the interplay between macroscopic quantum tunneling and nonlinearity-induced self-trapping of atoms in coupled BECs. The phenomenon of Bloch oscillations of these excitations is studied both for zero and nonzero backgrounds. We find that in a definite range of parameters, homogeneous distributions can become modulationally unstable. We also show that bright solitons and excitations of shock-wave type can exist in BEC arrays even in the case of positive scattering length. Finally, we argue that a BEC array with negative scattering length in the presence of linear potentials can display collapse.
We consider the three-dimensional (3D) mean-field model for the Bose–Einstein condensate, with a one-dimensional (1D) nonlinear lattice (NL), which periodically changes the sign of the nonlinearity along the axial direction, and the harmonic-oscillator trapping potential applied in the transverse plane. The lattice can be created as an optical or magnetic one, by means of available experimental techniques. The objective is to identify stable 3D solitons supported by the setting. Two methods are developed for this purpose: the variational approximation, formulated in the framework of the 3D Gross–Pitaevskii equation, and the 1D nonpolynomial Schrödinger equation (NPSE) in the axial direction, which allows one to predict the collapse in the framework of the 1D description. Results are summarized in the form of a stability region for the solitons in the plane of the NL strength and wavenumber. Both methods produce a similar form of the stability region. Unlike their counterparts supported by the NL in the 1D model with the cubic nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick may help to stabilize them against the collapse, by causing the solitons to shed the excess norm. A dynamical effect specific to the NL is found in the form of freely propagating small-amplitude wave packets emitted by perturbed solitons.
By employing a nonlinear three-mode model, we study the band structure of Bose-Einstein condensates in Fourier-synthesized optical lattices, where the nonlinearity comes from the mean field treatment of interaction between atoms. In the linear case, we present the band structure of the system. It is observed that the energy band structure is strongly dependent on the relative phase between the two lattice harmonics. In the nonlinear case, we show that the eigenenergy as a function of energy bias has a striking bowl structure in the middle energy level. It is found that there exist four critical values of interaction strength where the band structure undergoes interesting changes. Because of these topological distortions on the energy levels, we expect that the tunnelling dynamics make a dramatic change. The tunnelling probability is nonzero when the nonlinear term is very small and the energy levels keep the same topological structure as that in the linear case. Especially, the tunnelling probability as a function of the sweeping rate shows an irregular oscillation for the state on the middle level.