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Instabilities of a Bose-Einstein condensate with mixed nonlinear and linear lattices

Jun Hong, Chenhui Wang, and Yongping Zhang
Phys. Rev. E 107, 044219 – Published 27 April 2023
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  • INTRODUCTION
  • MODEL
  • NONLINEAR BLOCH BANDS AND ASSOCIATED…
  • INSTABILITIES OF NONLINEAR BLOCH STATES
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Bose-Einstein condensates (BECs) in periodic potentials generate interesting physics on the instabilities of Bloch states. The lowest-energy Bloch states of BECs in pure nonlinear lattices are dynamically and Landau unstable, which breaks down BEC superfluidity. In this paper we propose to use an out-of-phase linear lattice to stabilize them. The stabilization mechanism is revealed by the averaged interaction. We further incorporate a constant interaction into BECs with mixed nonlinear and linear lattices and reveal its effect on the instabilities of Bloch states in the lowest band.

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    • Received 23 January 2023
    • Accepted 13 April 2023

    DOI:https://doi.org/10.1103/PhysRevE.107.044219

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bose-Einstein condensatesCold gases in optical lattices
    Nonlinear DynamicsAtomic, Molecular & Optical

    Authors & Affiliations

    Jun Hong, Chenhui Wang, and Yongping Zhang*

    • Department of Physics, Shanghai University, Shanghai 200444, China

    • *yongping11@t.shu.edu.cn

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    Vol. 107, Iss. 4 — April 2023

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    • Figure 1
      Figure 1

      Nonlinear Bloch spectrum and associated nonlinear Bloch states of a BEC in a pure nonlinear lattice. g2=0.05 and V=0, g1=0. (a) The lowest two Bloch bands. (b),(c) The density distributions of nonlinear Bloch states at Brillouin zone center and edge in the lowest band, respectively [labeled by squares in panel (a)]. Pink stripes represent the regions that the nonlinear lattice is positive g2cosx>0.

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    • Figure 2
      Figure 2

      Nonlinear Bloch spectrum and associated nonlinear Bloch states of a BEC with a nonlinear lattice and an out-of-phase linear lattice. The nonlinear lattice amplitude is g2=0.05 and the constant interaction is g1=0. (a) The lowest two Bloch bands. When V<0.05 there is a gap opening between them (cyan-solid lines). V=0.05 is a critical value where the lowest two bands connect at Brillouin zone edges (black-solid lines). When V=0.08 gap is still closed (dotted lines). Further increasing V results in the gap reopening (red-solid lines). (b),(c) The density distributions of nonlinear Bloch states at Brillouin zone center (blue lines) and edge (red lines) in the lowest band for V=0.04 and V=0.12, respectively [labeled by squares in panel (a)]. Pink stripes represent the regions that the nonlinear lattice is positive g2cosx>0. Since the linear lattice is out-of-phase, in the striped regions the linear lattice is negative −Vcosx<0.

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    • Figure 3
      Figure 3

      Nonlinear Bloch spectrum and associated nonlinear Bloch states of a BEC with a nonlinear lattice and an in-phase linear lattice V<0. g2=0.05, V=−0.05, and g1=0. (a) The lowest two Bloch bands. (b),(c) The density distributions of nonlinear Bloch states at Brillouin zone center and edge in the lowest band, respectively [labeled by squares in panel (a)]. Dark-blue stripes represent the regions that the nonlinear lattice is positive g2cosx>0, since the linear lattice is in-phase, it is also positive −Vcosx>0 in striped regions.

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    • Figure 4
      Figure 4

      Instabilities of the BEC Bloch states in the lowest Bloch band with a nonlinear lattice and an out-of-phase linear lattice V>0. k and q are the quasimomenta of the Bloch states and perturbations, respectively. The colored shadow areas represent that the Bloch states are dynamical unstable, and the color scale labels the growth rate Γ defined in Eq. (9); the scale changes from the dark purple Γ=0 to bright red Γ=0.1. The gray areas indicate that the Bloch states have Landau instability. In the white regions, they are completely stable. For a fixed Bloch state represented by a fixed k, if there is any unstable mode in a q, the corresponding Bloch state is unstable.

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    • Figure 5
      Figure 5

      The averaged interaction G [defined in Eq. (11)] of the Bloch state at k=0 with a nonlinear lattice and an out-of-phase linear lattice. The horizontal red dashed line is G=0 for guiding eyes.

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    • Figure 6
      Figure 6

      Instabilities of the BEC Bloch states in the lowest Bloch band with a nonlinear lattice and an in-phase linear lattice V<0. The colored shadow areas represent that the Bloch states are dynamical unstable, and the color scale labels the growth rate Γ defined in Eq. (9); the scale changes from the dark purple Γ=0 to bright red Γ=0.1. The gray areas indicate that the Bloch states have Landau instability. In the white regions, they are completely stable.

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    • Figure 7
      Figure 7

      The dynamical-instability-phase-diagram of the BEC Bloch states at Brillouin zone center k=0 with the mixed nonlinear and linear lattices in the parameter space (g2,V). (a) g1=0, (b) g1=0.05, and (c) g1=−0.02. In the white regions, the k=0 Bloch state is dynamically stable; in the dark regions, the Bloch state is dynamically unstable. The red lines represent the zero averaged interaction G=0, and in the regions above the red lines the averaged interaction is repulsive and in the other regions it is attractive.

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    • Figure 8
      Figure 8

      The dynamical-instability-phase-diagram of the BEC Bloch states at Brillouin zone edge k=0.5 with the mixed nonlinear and linear lattices in the parameter space (g2,V). (a) g1=0, (b) g1=0.05, and (c) g1=−0.02. In the white regions, the k=0.5 Bloch state is dynamically stable; in the dark regions, the Bloch state is dynamically unstable.

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    • Figure 9
      Figure 9

      The time evolution of the k=0 Bloch states represented by the marked points in Fig. 7 with g1=0.05. Plots show the density distributions as a function of time t. (a) A stable evolution g2=0.1 and V=0.1. (b) An unstable evolution g2=0.15 and V=0.1.

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