Quantized transport of solitons in Bose-Einstein condensates driven by spin-orbit coupling

Abstract

We demonstrate that linear and nonlinear Thouless pumping can be realized in two-component elongated Bose-Einstein condensates using helicoidal spin-orbit coupling that slides with respect to a static optical lattice, identical for both spinor components. Stable quantized transport is found for solitons in semi-infinite and finite gaps, within certain intervals of chemical potentials and numbers of atoms. In the semi-infinite gap, the transport is arrested for solitons with sufficiently large number of atoms. We elucidate the important role of Zeeman splitting in the control of quantized transport, which disappears when the longitudinal component of the Zeeman field is removed.

Figures (5)

• Figure 1: Evolution of band edges of the Hamiltonian $H$ with time $t$ on one period $T$. Space‐time Chern numbers are indicated near corresponding bands. Hereafter we use the parameters $p=3$, $q=1$, $\alpha=6$, $V_0=6$, $\Delta_1=4$, and $\Delta_3=8$.
• Figure 2: Evolution dynamics (first column), wavepacket displacement $\delta x_c$ vs time $t$ (second column), and evolution of pseudospin components (third column) for excitation of the first (a) and third (b) bands in the linear lattice. Here and below $v=10^{-2}$. We show evolution of only the first component $|\Psi_1|$, since the behavior of $|\Psi_2|$ is qualitatively similar.
• Figure 3: (a) Families of solitons $N(\mu)$ in the semi-infinite and finite gap for attractive and repulsive interactions, respectively, and normalized one-cycle displacement, $\delta x_c(T)/X$, vs $\mu$. Vertical gray stripe indicates the first spectral band. Small squares with labels $a,b,c,d$ correspond to solitons whose transport is shown in Fig. \ref{['fig:solitons']}. (b) Ratio between IPRs of the initial soliton and of solution after one pumping cycle.
• Figure 4: Topological pumping of solitons with $\mu=-3.267$ (a), $\mu=-2.650$ (b), and $\mu=-2.240$ (c) for repulsive interactions, and soliton with $\mu=-3.750$ (d) for attractive interactions [see small squares in Fig. \ref{['fig:families']}(a) highlighting the locations of these solitons]. For each soliton, we plot the dynamics of the first component ($\Psi_1$) over two pumping cycles at $v=10^{-2}$, displacement of the center of mass, and projection of $\mathbf{\Psi }$ on Bloch bands at $t=0$ (black, red, and green lines correspond to projections on the first, second, and third bands, respectively). Only the dynamics of $|\Psi_1|$ is shown, as $|\Psi_2|$ behaves similarly.
• Figure 5: The one-cycle displacement for solitons (different $N$) and a Wannier function (WF) against the longitudinal Zeeman field strength $\Delta_1$.