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Formation of matter-wave soliton trains by modulational instability

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Science  28 Apr 2017:
Vol. 356, Issue 6336, pp. 422-426
DOI: 10.1126/science.aal3220
  • Fig. 1 Soliton-train formation from modulational instability (MI).

    (A) Schematic representation of the effects of a scattering length quench. At short times, the condensate has not responded to changes in scattering length. MI results in rapid growth of fluctuations at a length scale of 2πξ. Atoms flow toward regions of high density on a time scale of γ–1, owing to a nonlinear focusing from attractive interactions. Solitons are formed for t > γ–1. (B) Column density images for af = –0.18a0. Immediately after the quench, there is no discernible change in Na, nor is there any change in shape from that of the original condensate at ai = +3a0. Solitons form at later times and undergo breathing and dipole oscillations. (C) Similar to (B), except with af = –2.5a0. Modulations appear much earlier, as do gaps near the center where the density of the original condensate was high, which we attribute to primary collapses. A reduction in Ns is evident at longer th. Each image corresponds to a different experimental run, and hence, real-time dynamics cannot be directly inferred from these images. Here, z is the position along the axial coordinate.

  • Fig. 2 Postquench evolution of atom number.

    (A) Na versus th for various af. The arrows indicate the calculated γ–1 for each value of af, which is determined using the peak value of n1D. The black dashed line corresponds to half of a breathing period (tbr = 68 ms). We observe a plateau in Na for each af, followed by a rapid decrease in atoms starting shortly after th ≈ γ–1. We attribute the lack of a plateau for af = –2.5a0 to tr > γ–1. (B) Data replotted versus thγ. The data collapse onto a single curve, except for af = –2.5a0. The data are fit to a power law, Embedded Image, shown as a solid black line, where κ = –0.35(1) for both fits. For all af, points for th > tbr/2 have been omitted from the fit. Error bars are the SD of the mean of up to 30 shots.

  • Fig. 3 Postquench evolution of soliton number and strength of nonlinearity.

    (A) Ns versus af. The dashed line corresponds to a fit of the data to the model (see text), where an overall scaling of 1.04(2) is the only fit parameter. Data for Embedded Image are omitted from the fit. We attribute the suppression in Ns for Embedded Image to primary collapse, resulting in a reduction in the number of solitons formed. (B) Ns versus th. Ns does not change with th for the two smallest Embedded Image, whereas for larger Embedded Image, Ns decays with th. Dashed lines correspond to the initial number of solitons. (C) Δ versus th. The initial value of Δ = Na/(NsNc) increases as Embedded Image is increased and is consistent with an expected Embedded Image scaling. This trend continues up to Δ = 1, above which the solitons are unstable against primary collapse. Error bars are the SD of the mean of up to 30 shots.

  • Fig. 4 Soliton-train dynamics.

    (A) Multiple images of the same soliton train, for af = –0.18a0. Beginning at th = 10 ms, a new image was taken every 2 ms. We infer dominantly repulsive interactions, although occasional attractive collisions occur between neighbors. The reduction in the overall size of the train is caused by a breathing mode excited by the quench, and a dipole oscillation is also evident. (B) Similar to (A), starting with th = 40 ms. The effects of the breathing mode in its expansion phase are evident.

Supplementary Materials

  • Formation of matter-wave soliton trains by modulational instability

    Jason H. V. Nguyen, De Luo, Randall G. Hulet

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