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Nonlinear waves in an experimentally motivated ring-shaped Bose-Einstein-condensate setup

M. Haberichter, P. G. Kevrekidis, R. Carretero-González, and M. Edwards
Phys. Rev. A 99, 053619 – Published 29 May 2019
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  • INTRODUCTION
  • MODEL AND METHODOLOGY
  • NUMERICAL SIMULATIONS
  • CONCLUSIONS AND FUTURE CHALLENGES
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We systematically construct stationary soliton states in a one-component, two-dimensional, repulsive, Gross-Pitaevskii equation with a ring-shaped targetlike trap similar to the potential used to confine a Bose-Einstein condensate in a recent experiment [R. Mathew, A. Kumar, S. Eckel, F. Jendrzejewski, G. K. Campbell, M. Edwards, and E. Tiesinga, Phys. Rev. A 92, 033602 (2015)]. In addition to the ground-state configuration, we identify a wide variety of excited states involving phase jumps (and associated dark solitons) inside the ring. These configurations are obtained from a systematic bifurcation analysis starting from the linear, small atom density, limit. We study the stability and, when unstable, the dynamics of the most basic configurations. Often these lead to vortical dynamics inside the ring persisting over long time scales in our numerical experiments. To illustrate the relevance of the identified states, we showcase how such dark-soliton configurations (even the unstable ones) can be created in laboratory condensates by using phase-imprinting techniques.

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    • Received 12 January 2019

    DOI:https://doi.org/10.1103/PhysRevA.99.053619

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bose-Einstein condensates
    1. Physical Systems
    Solitons
    Atomic, Molecular & OpticalNonlinear Dynamics

    Authors & Affiliations

    M. Haberichter1,*, P. G. Kevrekidis1, R. Carretero-González2, and M. Edwards3,4,†

    • 1Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
    • 2Nonlinear Dynamical Systems Group,‡ Computational Sciences Research Center, and Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182-7720, USA
    • 3Department of Physics, Georgia Southern University, Statesboro, Georgia 30460-8031, USA
    • 4Joint Quantum Institute, National Institute of Standards and Technology and the University of Maryland, Gaithersburg, Maryland 20899, USA

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    Vol. 99, Iss. 5 — May 2019

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    Images

    • Figure 1
      Figure 1

      Ring-shaped trapping potential V, given in Eq. (3), corresponding to an experiment performed at NIST [56]. In this figure, and all subsequent ones, space (x,y) is displayed using physical units (in microns).

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    • Figure 2
      Figure 2

      Effects of the disk population on the ring population for the indicated values of μ. (a) Central slice (y=0) for the ground-state density |u(x)|2 for the full potential V(r) in Eq. (3). (b) Slice for the “ring” ground state |uring(x)|2 corresponding to the case when the central (disk) portion of the potential is removed. (c, d) Density difference between the full ground state and the ring ground state Δ(x)≡|u(x)|2−|uring(x)|2. Note that the ring portion for the difference is at most of the order 10−4 when μ=0.8.

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    • Figure 3
      Figure 3

      Ground-state and n-dark soliton solutions for μ=0.9. The real part and density of the solutions are depicted, respectively, in the top and bottom rows of panels. (a) Ground state (that populates the central well of the external potential). (b) Basic ring state without any dark solitons. (c–e) First three excited states along the ring containing, respectively, two, four, and six dark solitons. All these stationary solutions are purely real.

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    • Figure 4
      Figure 4

      Particle number N as a function of μ for the ground state (GS) and the n-soliton (nS) stationary steady states. These steady states are obtained by continuation from the N≃0 limit where the solutions are calculated by taking an initial guess in our fixed-point iterations corresponding to the first few eigenfunctions (excited states) in the linear limit. The critical chemical potential values μcrit at which the different states are found to emerge correspond to μcrit(0)=0.313 for 0S, μcrit(2)=0.314 for 2S, μcrit(4)=0.316 for 4S, μcrit(6)=0.320 for 6S, and μcrit(8)=0.326 for 8S. The corresponding profiles for these solutions for μ=0.9 are depicted in Fig. 3.

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    • Figure 5
      Figure 5

      Double-ring solution and some of its bifurcating states for μ=0.7. (a) Double-ring solution (that bifurcated from the ground state) consisting of two concentric out-of-phase rings. (b–d) Successive states bifurcating away from the double-ring solution. The corresponding particle numbers for these solutions as a function of μ are depicted in Fig. 6. The layout is the same as in Fig. 3.

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    • Figure 6
      Figure 6

      Particle number N as a function of μ for the ground state (GS), the double-ring (a), and its first three bifurcating branches (b–d). The corresponding profiles for μ=0.7 are depicted in Fig. 5. The double ring bifurcates from the ground state at μ≃0.560, while the subsequent states bifurcate in turn from the double-ring solution for (b) μ=0.586, (c) 0.618, and (d) 0.708.

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    • Figure 7
      Figure 7

      Stability BdG spectra for the ground state (top row of panels) and the double-ring state (bottom row of panels) as a function of the chemical potential μ. The corresponding profiles are depicted in the first row of panels of, respectively, Figs. 3 and 5. The left and right panel depict, respectively, the real and imaginary parts of the spectra. Recall that (neutral) stability is only achieved when Im(ω)=0. The ground state is always (neutrally) stable while the double-ring state is, since its inception, always unstable.

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    • Figure 8
      Figure 8

      Evolution of the double-ring configuration [see Fig. 5] slightly perturbed (10−3 times the normalized eigenvector) with an eigenvector picked from the third instability in the BdG spectra (see bottom panels in Fig. 7). The top, middle, and bottom rows of panels display, respectively, the real part, the density, and the phase of the profiles at the times indicated. In this figure, as is the case in all the figures in this paper, the indicated times are measured in nondimensional units as per the adimensionalization discussed below Eq. (2).

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    • Figure 9
      Figure 9

      Two-dark soliton profile and its bifurcating states for μ=0.7. The layout is the same as in previous figures. The corresponding particle numbers as a function of μ are depicted in Fig. 10.

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    • Figure 10
      Figure 10

      Top: Particle number N as a function of μ for the stationary states bifurcating from the two-soliton solution (a). Bottom: Particle number difference ΔN between the states bifurcating from the two-soliton configuration and the two-soliton configuration itself. The corresponding profiles are depicted in Fig. 9. The first three bifurcating states from the two-soliton solution (a) bifurcate at (b) μ≃0.321, (c) μ≃0.466, and (d) μ≃0.614.

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    • Figure 11
      Figure 11

      Same as Fig. 9, but showing (from left to right) the first state bifurcating from the four-, six-, and eight-soliton profiles.

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    • Figure 12
      Figure 12

      Stability BdG spectra for the two-soliton configuration and its first bifurcating state as a function of μ. The layout is the same as in Fig. 7. The corresponding profiles for μ=0.7 are depicted, respectively, in the first two columns of Fig. 9.

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    • Figure 13
      Figure 13

      Density (top row of panels) and phase (bottom row of panels) plots showing the time evolution of the two-soliton ground state heavily perturbed (ten times the normalized eigenvector) with an eigenvector picked from the first instability in the BdG spectra (see Fig. 12).

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    • Figure 14
      Figure 14

      Density (top row of panels) and phase (bottom row of panels) plots showing the time evolution of the two-soliton ground state heavily perturbed (30 times the normalized eigenvector) with an eigenvector picked from the second instability in the BdG spectra (see Fig. 12). The evolution past the times shown (t>60) is qualitatively similar to the one depicted in Fig. 13.

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    • Figure 15
      Figure 15

      Stability BdG spectra for the four-, six-, and eight-soliton states (from top to bottom). The layout is the same as in Fig. 7. The corresponding profiles are depicted in Figs. 3–3.

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    • Figure 16
      Figure 16

      Evolution dynamics for the four-soliton (top row), six-soliton (middle row), and eight-soliton (bottom row) configurations heavily perturbed (30 times the normalized eigenvector) with an eigenvector picked from the first instability of the corresponding BdG spectra for μ=0.7.

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    • Figure 17
      Figure 17

      Phase imprinting in the case of stable two-soliton (top) and four-soliton (bottom) configurations for μ=0.6. The corresponding top, middle, and bottom series of panels depict, respectively, the real part, the density, and the phase of the solution. One can observe that the configurations robustly persist in the condensate dynamics.

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    • Figure 18
      Figure 18

      Dynamics ensuing from the phase imprinting of the two-soliton (top) and four-soliton (bottom) configurations for μ=0.9. For this value of the chemical potential the corresponding n-soliton steady states are unstable and thus evolve in a manner akin to what is shown in Fig. 16.

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    • Figure 19
      Figure 19

      Real states bifurcating from the ground state for μ=0.9 [except μ=0.96 for panel (k)].

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    • Figure 20
      Figure 20

      Real states calculated from the dipole state for μ=0.9

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    • Figure 21
      Figure 21

      (a–d) Complex states calculated from the ground state. These profiles correspond to an n-dark soliton state coupled to a the ground state. (e–h) Complex states calculated from the dipole state. These profiles correspond to an n-dark soliton state coupled to a dipole state at the center of the cloud. (i–k) Vortexlike states calculated from the ground state. These states are similar to the ones depicted in panels (a)–(d) by replacing the n-dark soliton state by a ring of n vortices. μ=0.9 in all cases.

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