Collisionally Inhomogeneous Bose-Einstein Condensates with a Linear Interaction Gradient

We study the evolution of a collisionally inhomogeneous matter wave in a spatial gradient of the interaction strength. Starting with a Bose-Einstein condensate with weak repulsive interactions in quasi-one-dimensional geometry, we monitor the evolution of a matter wave that simultaneously extends into spatial regions with attractive and repulsive interactions. We observe the formation and the decay of solitonlike density peaks, counterpropagating self-interfering wave packets, and the creation of cascades of solitons. The matter-wave dynamics is well reproduced in numerical simulations based on the nonpolynomial Schrödinger equation with three-body loss, allowing us to better understand the underlying behavior based on a wavelet transformation. Our analysis provides new understanding of collapse processes for solitons, and opens interesting connections to other nonlinear instabilities.

Figure 1

(a) Experimental setup with coils and with the vertical and horizontal laser beams $L_V$ and $L_H$. The gravitational force, $F_{\mathrm{grav}}$, is canceled by the magnetic force $F_{\mathrm{mag}}$. (b) Position dependence of the scattering length (left axis) and the magnetic field (right axis). Inset: absorption image of the initial density distribution at position $z=0\mu \mathrm{m}$ after a free expansion time of 25 ms. (c) Absorption images of the density distribution after a variable evolution time $t$ and an expansion time of 25 ms. The color map of each image is rescaled using the peak density to enhance the visibility of the evolving structures in the density profiles. (d)–(g) Horizontally integrated density profiles of the images in (c).

Figure 2

(a) Calculated density profile of the ground state for $N=13000$, ${\omega }_r=2\pi \times 40\mathrm{Hz}$, ${\omega }_z=2\pi \times 5.2\mathrm{Hz}$, $a=3a_0$ (red) and $a_{\mathrm{off}}=36a_0$ (blue) with ${\partial }_{z}a=0.21a_0/\mu \mathrm{m}$ (solid lines) and ${\partial }_{z}a=0a_0/\mu \mathrm{m}$ (dashed lines). (b) Absorption images and (c) 1D-density profiles of a BEC for varying hold times $t$ and a constant free expansion time of 20 ms. $a_{\mathrm{off}}$ is $36a_0$ at the initial position of BEC. (d) Illustration of forces on sections of a BEC and soliton, and (e) during the expansion into regions with attractive interaction. Red (blue) patches and arrows indicate sections of the matter wave that are pushed toward smaller (larger) scattering length.

Figure 3

(a) Numerical simulation of the evolution of the density profile using NPSE with three-body loss (1D density in $\mathrm{atoms}/\mu \mathrm{m}$). Inset: Zoomed region of the growing and then decaying soliton. (b) Wavelet decomposition [20] of the atomic wave function at $t=150\mathrm{ms}$ [first vertical line in (a)], showing the emerging soliton (white arrow). (c) Wavelet decomposition at $t=190\mathrm{ms}$ [second vertical line in (a)], after the appearance of the density ripples. The arrow indicates the counterpropagating wave packet. (d) Wavelet decomposition at $t=260\mathrm{ms}$ [third vertical line in (a)], after the counterpropagating wave packet identified by the arrow penetrates the region of repulsive interactions ($a>0$).

Figure 4

Ripple pattern in the density profile for approximately 22000 atoms. (a) Absorption image of density profile after a free expansion time of 25 ms. (b) Positions of density peaks for multiple absorption images. Mean position and standard deviation of the numbered peaks are indicated by dashed lines and by gray areas, respectively. (c) Simulated density distribution at $\mathrm{\Delta }T=40\mathrm{ms}$ after the decay of the soliton. (d) Measured density distribution at $\mathrm{\Delta }T=40\mathrm{ms}$ [20].