Interaction-Tuned Dynamical Transitions in a Rashba Spin-Orbit-Coupled Fermi Gas

We consider the time evolution of the magnetization in a Rashba spin-orbit coupled Fermi gas, starting from a fully polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting nonlinear system of equations gives rise to three distinct dynamical regimes with qualitatively different asymptotic behaviors of the magnetization at long times. The distinct regimes and the transitions between them are controlled by the ratio of interaction and spin-orbit coupling strength $\lambda$: for small $\lambda$, the magnetization decays to zero. For intermediate $\lambda$, it displays undamped oscillations about zero, and for large $\lambda$, a partially magnetized state is dynamically stabilized. The dynamics we find is a spin analog of interaction induced self-trapping in double-well Bose Einstein condensates. The predicted phenomena can be realized in trapped Fermi gases with synthetic spin-orbit interactions.

Figure 1

Schematic plot showing nonequilibrium steady states of an initially spin-polarized Fermi gas in the presence of SOC. Here, $\lambda =g{n}_0/\alpha p_{\mathrm{th}}$, where $g{n}_0$ parametrizes the interactions [Eq. (2)] and $\alpha p_{\mathrm{th}}$ parametrizes the SOC [Eq. (3)]. Different types of steady states are separated by transition points ${\lambda }_{c1}$ and ${\lambda }_{c2}$. For small interactions ($\lambda <{\lambda }_{c1}$), magnetization decays to zero. For ${\lambda }_{c1}<\lambda <{\lambda }_{c2}$, magnetization oscillates forever around zero, while for large interactions ($\lambda >{\lambda }_{c2}$), the system becomes partially polarized.

Figure 2

(Top, left) SOC and interactions can be viewed as magnetic fields in the longitudinal (red solid arrow) and transverse direction (blue solid arrow), respectively. The spin precesses around ${\mathit{B}}_{\mathrm{eff}}={\mathit{B}}_{\mathrm{SOC}}+{\mathit{B}}_{\mathrm{MF}}$. (Right panels) Snapshots of the magnetization for various interaction strengths showing the three different final states: unpolarized (UP), oscillating magnetization (OM), and partially polarized (PP) state. (Top) Solid red curve is the noninteracting result (see text for analytic formula), and the dashed-dotted blue curve has $\lambda =1.5$. (Center) $\lambda =2.3$ (solid red line) and the strongly anharmonic curve is $\lambda =2.977$. (Bottom) $\lambda =4$. (Bottom, left) The long time-averaged magnetization (dashed blue) and the amplitude of fluctuations about the average $\delta m_z=\sqrt{⟨{(m_z-⟨m_z⟩)}^2⟩}$ (solid red), for different values of $\lambda$. At a first critical value, ${\lambda }_{c1}=1.533(3)$, the magnetization develops undamped oscillations about zero, and at a second critical value ${\lambda }_{c2}=2.9825(2)$, the net magnetization jumps to a nonzero at long times (vertical blue dotted line).

Figure 3

Our spin system as a collection of double wells, corresponding to spin-$\uparrow$ (blue oval, left well) and spin-$\downarrow$ (red oval, right well) at each momentum. The intrawell tunneling is provided by the spin-orbit magnetic field $B_{\mathrm{SOC}}(\mathit{p})$ (red arrow), which is momentum dependent, while the different wells are coupled to one another via the interaction term $B_{\mathrm{MF}}(t)$ (blue arrow). Arrow thickness denotes the relative strength of interactions and SOC. (Left) Weak interactions ($B_{\mathrm{MF}}\ll B_{\mathrm{SOC}}$), (Right) strong interactions ($B_{\mathrm{MF}}\gg B_{\mathrm{SOC}}$). See the text for details.