Coherent spin mixing via spin-orbit coupling in Bose gases

We study beyond-mean-field properties of interacting spin-1 Bose gases with synthetic Rashba-Dresselhaus spin-orbit coupling at low energies. We derive a many-body Hamiltonian following a tight-binding approximation in quasimomentum space, where the effective spin dependence of the collisions that emerge from spin-orbit coupling leads to dominant correlated tunneling processes that couple the different bound states. We discuss the properties of the spectrum of the derived Hamiltonian and its experimental signatures. In a certain region of the parameter space, the system becomes integrable, and its dynamics becomes analogous to that of a spin-1 condensate with spin-dependent collisions. Remarkably, we find that such dynamics can be observed in existing experimental setups through quench experiments that are robust against magnetic fluctuations.

Figure 1

Emergence of spin-changing collisions in the presence of SOC. (a) Dispersion bands of Hamiltonian (1) along the longitudinal direction $\widehat{z}$ for ${\mathrm{\Delta }}_s=0$, and $\hslash \mathrm{\Omega }=0.2E_R$. The color texture represents the expected value of the spin of the dressed states. Dashed lines show the undressed dispersion bands (at $\mathrm{\Omega }=0$). (b) Schematic representation of a resonant collision process that couples different spin states mediated by Raman transitions (represented in wavy lines). (c) Spin-changing collision processes that couple the many-body states in the Fock space spanned by the tight-binding well-states basis $|L\rangle ,|M\rangle$, and $|R\rangle$ (see Sec. 3), which act as effective correlated tunneling processes between the bound states.

Figure 2

Spectrum of the tight-binding Hamiltonian in the nonintegrable regime: eigenvalues ${\lambda }_i$ of (34) with $\epsilon =0$ and $U_2=0$, in the even-parity subspace for (a) $N=2$ and (b) $N=4$, as a function of $\chi =\frac{2}{\pi }arctan\left(\frac{U_1}{J}\right)$. The gaps at the avoided crossings are small but nonzero, as illustrated in the inset. The energy gap between the two lowest energy eigenstates is plotted in (c), for $N=4$ (solid blue), $N=10$ (dashed red), and $N=14$ (dotted black).

Figure 3

Spectrum of the tight-binding Hamiltonian in the integrable regime: eigenvalues ${\lambda }_i$ of (34) with $\epsilon =U_1/2$ and $U_2=0$, in the even-parity subspace for (a) $N=2$ and (b) $N=4$, as a function of $\chi =\frac{2}{\pi }arctan\left(\frac{U_1}{J}\right)$. The gaps at the crossings vanish at this value of the parameter $\epsilon$. The energy gap between the two lowest energy eigenstates is plotted in (c) for $N=4$ (solid blue), $N=10$ (dashed red), and $N=14$ (dotted black).

Figure 4

Spin mixing induced by effective spin-dependent collisions. Mean population in $|M\rangle$ as a function of time for a state initially prepared within the $m_z=0$ subspace at $|0N0\rangle$ (dashed blue), $|\frac{N}{3}\frac{N}{3}\frac{N}{3}\rangle$ (solid green), and $|\frac{N}{2}0\frac{N}{2}\rangle$ (dashed-dotted red). The initial state is evolved under Hamiltonian (34) for $J=U_2=0$ and $\epsilon =U_1/2$. Time is scaled to $\hslash /U_1$.

Figure 5

Signature of the spin-mixing dynamics. (a) Mean relative population in the edge wells, $\langle \widehat{N_L}+\widehat{N_R}\rangle$, as a function of time for a state initially prepared at $|0N0\rangle$, with $N=50$, and evolved under Hamiltonian (34) for $J=U_2=0$ and $\epsilon =-10U_1$ (dashed green), $\epsilon =0.5U_1$ (solid blue), and $\epsilon =10U_1$ (dashed-dotted red). Time is scaled to $\hslash /U_1$. (b) Time-averaged relative value of $\langle \widehat{N_L}+\widehat{N_R}\rangle$, as a function of $\epsilon$, averaged over large times, $\mathrm{\Delta }t\sim 1/U_1$, for different numbers of particles: $N=5$ (solid red), $N=10$ (dashed orange), $N=45$ (dotted blue), and $N=50$ (dashed-dotted purple). (c) Width $\mathrm{\Delta }\epsilon$ of the peak as a function of the total number of particles $N$ for $\alpha =0.1$ with optimal parameters and considering ${}^{87}\mathrm{Rb}$ atoms. $\hslash \mathrm{\Omega }/E_R$ is set to 0.5, 0.3, and 0.1, plotted in squared blue, circular purple, and triangular green dots, respectively.