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arXiv:1605.02984 · Physical Review A

Nonlinear looped band structure of Bose-Einstein condensates in an optical lattice

Silvio Koller, Elizabeth A. Goldschmidt, Roger C. Brown, Robert Wyllie, Ryan Wilson, J. V. Porto

Published Dec 23, 2016• 22 citations
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Abstract

We study experimentally the stability of excited, interacting states of bosons in a double-well optical lattice in regimes where the nonlinear interactions are expected to induce ``swallowtail'' looped band structure. A dynamic homogeneous Gross-Pitaevskii calculation indicates that the double-well lattice both stabilizes the looped band structure and allows for dynamic preparation of different initial states, including states within the loop structure. The homogeneous calculation predicts that the loop states, unlike the ground states, should be dynamically stable. An inhomogeneous mean-field calculation including the trap potential, however, implies that the decay is dominated by inhomogeneous effects and that there is little variation in the decay rate among the states prepared within the loop structure. By experimentally preparing different initial coherent states and observing their subsequent decay, we observe distinct decay rates in regimes where multivalued looped band structure is expected, although not the stability predicted by the homogeneous calculation.

DOI
10.1103/physreva.94.063634

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We study experimentally the stability of excited, interacting states of bosons in a double-well optical lattice in regimes where the nonlinear interactions are expected to induce ``swallowtail'' looped band structure. A dynamic homogeneous Gross-Pitaevskii calculation indicates that the double-well lattice both stabilizes the looped band structure and allows for dynamic preparation of different initial states, including states within the loop structure. The homogeneous calculation predicts that the loop states, unlike the ground states, should be dynamically stable. An inhomogeneous mean-field calculation including the trap potential, however, implies that the decay is dominated by inhomogeneous effects and that there is little variation in the decay rate among the states prepared within the loop structure. By experimentally preparing different initial coherent states and observing their subsequent decay, we observe distinct decay rates in regimes where multivalued looped band structure is expected, although not the stability predicted by the homogeneous calculation.
In this work, we consider excited many-body mean-field states of bosons in a double-well optical lattice by investigating stationary Bloch solutions to the nonlinear equations of motion. We show that, for any positive interaction strength, a loop structure emerges at the edge of the band structure whose existence is entirely due to interactions. This can be contrasted to the case of a conventional optical (Bravais) lattice where a loop appears only above a critical repulsive interaction strength. Motivated by the possibility of realizing such nonlinear Bloch states experimentally, we analyze the collective excitations about these nonlinear stationary states and thereby establish conditions for the system's energetic and dynamical stability. We find that there are regimes that are dynamically stable and thus apt to be realized experimentally.
We calculate the band structure of ultracold atoms located inside a laser-driven optical cavity. For parameters where the atom-cavity system exhibits bistability, the atomic band structure develops loop structures akin to the ones predicted for Bose-Einstein condensates in ordinary (noncavity) optical lattices. However, in our case the nonlinearity derives from the cavity back-action rather than from direct interatomic interactions. We find both bi- and tristable regimes associated with the lowest band, and show that the multistability we observe can be analyzed in terms of swallowtail catastrophes. Dynamic and energetic stability of the mean-field solutions is also discussed, and we show that the bistable solutions have, as expected, one unstable and two stable branches. The presence of loops in the atomic band structure has important implications for proposals concerning Bloch oscillations of atoms inside optical cavities [Peden et al., Phys. Rev. A 80, 043803 (2009); Prasanna Venkatesh et al., Phys. Rev. A 80, 063834 (2009)].
The band structure of a Bose-Einstein condensate is studied for lattice traps of sinusoidal, Jacobi elliptic, and Kronig-Penney form, all in the context of the nonlinear Schr\"odinger equation. It is demonstrated that the physical properties of the system are independent of the type of lattice. The Kronig-Penney potential, which admits a full exact solution in closed analytical form, is then used to understand the swallowtails, or loops, that form in the band structure. The appearance of swallowtails is explained by adiabatically tuning a second lattice with half the period. Such a two-color lattice, which can be easily realized in experiments, has intriguing physical properties. For instance, swallowtails appear even for weak nonlinearity, which is the experimental regime. We determine the stability properties of this system and relate them to current experiments.
Energy band structure and stability of Bose-Einstein condensate with both two- and three-body interactions in one-dimensional periodic optical lattices are discussed. It is demonstrated that the lowest energy band can show a loop structure at the boundary of the first Brillouin zone only when the number of atoms is larger than a critical value. When the loop structure is excited, an analytical expression of the loop width is obtained. The energetical and dynamical stabilities of the system are determined by considering the behavior of the Gross-Pitaevskii energy functional when the condensate wave function diverge away from a stationary state upon small perturbation, and its dependence on the number of atoms, the magnitude of the periodic potential, especially, the strength of the three-body force are studied. In addition, a criteria of the stability for excitations with long wavelengths is given by applying the hydrodynamic approach. The influence of the three-body interaction on the band properties and the stabilities are discussed in detail. It is shown that the critical density for developing the loop structure, the loop width, and the instability criteria are modified dramatically by the three-body interaction.
All Bloch states of the mean field of a Bose-Einstein condensate in the presence of a one dimensional lattice of impurities are presented in closed analytic form. The band structure is investigated by analyzing the stationary states of the nonlinear Schr\odinger, or Gross-Pitaevskii, equation for both repulsive and attractive condensates. The appearance of swallowtails in the bands is examined and interpreted in terms of the condensates superfluid properties. The nonlinear stability properties of the Bloch states are described and the stable regions of the bands and swallowtails are mapped out. We find that the Kronig-Penney potential has the same properties as a sinusoidal potential; Bose-Einstein condensates are trapped in sinusoidal optical lattices. The Kronig-Penney potential has the advantage of being analytically tractable, unlike the sinusoidal potential, and, therefore, serves as a good model for experimental phenomena.
All Bloch states of the mean field of a Bose-Einstein condensate in the presence of a one-dimensional lattice are presented in closed analytic form. The band structure is investigated by analyzing the stationary states of the nonlinear Schr\"odinger, or Gross-Pitaevskii, equation for both repulsive and attractive condensates. The appearance of swallowtails in the bands is examined and interpreted in terms of the condensates superfluid properties. The nonlinear stability properties of the Bloch states are described and the stable regions of the bands and swallowtails are mapped out. We find that the Kronig-Penney potential has the same properties as a sinusoidal potential. Bose-Einstein condensates are trapped in sinusoidal optical lattices. The Kronig-Penney potential has the advantage of being analytically tractable, unlike the sinusoidal potential, and, therefore, serves as a good model for understanding experimental phenomena.