Cavity-assisted dynamical spin-orbit coupling in cold atoms

We consider ultracold atoms subjected to a cavity-assisted two-photon Raman transition. The Raman coupling gives rise to effective spin-orbit interaction which couples the atom's center-of-mass motion to its pseudospin degrees of freedom. Meanwhile, the cavity photon field is dynamically affected by the atom. This feedback between the atom and photons leads to a dramatic modification of the atomic dispersion relation, and further leads to dynamical instability of the system. We propose to detect the change in the cavity photon number as a direct way to demonstrate dynamical instability.

Figure 1

(a) Schematic diagram of the cavity-assisted spin-orbit coupled system. (b) Level diagram of atom and light field configuration.

Figure 2

Eigenenergy $\epsilon$ as a function of quasimomentum. We set $\tilde{\delta }=0$ and ${\varepsilon }_p=\kappa$. For nonzero ${\delta }_c$, a loop structure forms when ${\Omega }_c^{\left(1\right)}<\Omega <{\Omega }_c^{\left(2\right)}$. For ${\delta }_c=\pm \kappa$, ${\Omega }_c^{\left(1\right)}=4{\varepsilon }_p$ and ${\Omega }_c^{\left(2\right)}=4\sqrt{2}{\varepsilon }_p$. Throughout our calculation, we take $\kappa$ and $\sqrt{2m\kappa }$ to be the units for energy and momentum, respectively. A typical value for $\kappa$ is $2\pi \times 1$ MHz, and we choose $q_r=0.22$ in our units.

Figure 3

Photon number distribution as a function of the atom's quasimomentum. The parameters are the same as in the right column of Fig. 2, where ${\delta }_c=\kappa$, ${\varepsilon }_p=\kappa$, and $\Omega /\kappa =3$, 5, and 7 from (a) to (c).

Figure 4

Stability analysis of the dispersion curve. Colored triangles represent dynamically unstable states and black solid dots represent dynamically stable ones. The color bar represents $\gamma$, the decay rate of the unstable states. In all figures, $\Omega =1.1\kappa$ and ${\delta }_c=\kappa$. From (a) to (c) ${\varepsilon }_p=2\kappa$ and $\tilde{\delta }=0.05$, 0, and $-0.05\kappa$; from (d) to (f) ${\varepsilon }_p=0.2\kappa$, and $\tilde{\delta }=0.05$, 0, and $-0.05\kappa$.

Figure 5

Evolution of photon number. The initial states are prepared using the same set of parameters as in Fig. 4. In (a), we start from point $P_1$ with $k_z=-2q_r$ and in (b) we start from point $P_2$ with $k_z=0$, both indicated in Fig. 4. From $t=0$ to $4000/\kappa$, $\tilde{\delta }$ is linearly tuned from $0.05\kappa$ to $-0.05\kappa$ and remains fixed afterwards. Red solid dots represent the photon number corresponding to the instantaneous eigenstate, while blue solid lines represent the dynamical evolution according to Eqs. (3) and (4) after mean-field approximation.