Anderson Localization of Solitons

At low temperature, a quasi-one-dimensional ensemble of atoms with an attractive interaction forms a bright soliton. When exposed to a weak and smooth external potential, the shape of the soliton is hardly modified, but its center-of-mass motion is affected. We show that in a spatially correlated disordered potential, the quantum motion of a bright soliton displays Anderson localization. The localization length can be much larger than the soliton size and could be observed experimentally.

Figure 1

Inverse localization length of a bright soliton versus its momentum $k$ in units of its width $\xi =0.02$ ($\approx 1\mu \mathrm{m}$) in a speckle potential with correlation length ${\sigma }_0=0.28\xi$. Circles are numerical results obtained by diagonalizing the effective Hamiltonian (3), solid lines are numerical results from a transfer-matrix calculation. The bare potential amplitude is $V_0=\pm 0.1$ [blue (dark gray) and red (gray) curve] in our units, corresponding to $\pm 8\times {10}^{-5}|\mu |$. Inset: same data in logarithmic scale. The lowest-order perturbation result, Eq. (9) is shown as a black dashed line.

Figure 2

Probability distribution of soliton position $q$ in units of its width $\xi =0.02$ ($\approx 1\mu \mathrm{m}$). Blue (dark gray) and red (gray) lines: Data from numerical integration of the Schrödinger equation with the Hamiltonian (3) starting in the ground state of an axial harmonic trap with ${\omega }_z=100$ ($2\pi \times 64\mathrm{Hz}$) and lasting $\Delta t=20$ (5 s) for a blue (dark gray) and red (gray)-detuned speckle potential of amplitude $V_0=\pm 8\times {10}^{-5}\mu$ ($2\pi \times 0.064\mathrm{Hz}$). Data averaged over 250 different realizations of the disorder potential. Black dashed line: saddle-point result (13) for parameters in the Born approximation. The full integral (12) yields the same curve for $|q|\gg \xi$. The double-logarithmic inset plot shows the dominantly algebraic decrease as $|q{|}^{-1}$.