Spin evolution of cold atomic gases in $\mathrm{SU}\left(2\right)\hspace{0.17em}\otimes \hspace{0.17em}\mathrm{U}\left(1\right)$ fields

We consider response function and spin evolution in spin-orbit coupled cold atomic gases in a synthetic gauge magnetic field influencing solely the orbital motion of atoms. We demonstrate that various regimes of spin-orbit coupling strength, magnetic field, and disorder can be treated within a single approach based on the representation of atomic motion in terms of auxiliary collective classical trajectories. Our approach allows a unified description of fermionic and bosonic gases.

Figure 1

Synthetic magnetic field $\mathbf{B}$, the spin-orbit coupling momentum $\mathbf{q}$, and the corresponding spin-split Fermi surfaces with the Fermi momentum $p_F.$ Vector $\mathbf{h}$ is the direction of the spin-orbit coupling field in Eq. (3) and $\mathbf{\sigma }$ shows the equilibrium magnetization of the components of the spin-split Fermi gas. This picture corresponds to ultrastrong coupling with $q>p_F.$

Figure 2

Dependence of the dimensionless ratio $S\left(t\right)/S\left(0\right)$ for the total spin calculated with exact Eqs. (4) and (27) on the dimensionless product ${\omega }_{c}t$ for different angles $\theta$ (see Fig. 1) as marked near the plots. (a) For fermions with the Fermi momentum $p_F$, spin-orbit coupling $q=p_F/2,$ and cyclotron ${\omega }_c=E_F/8.$ (b) For bosons with the momentum $p_{\mu },$ spin-orbit coupling $q=p_{\mu }/2,$ and ${\omega }_c=\left|\mu \right|/8.$ An approximation with Eq. (28) (not shown here) describes all evolutions very accurately. The insets show the time dependences for $\theta =\pi /4.$