Fig. 1. (a) Schematic for creating the two-component TCM. (b) Density distribution of self-ordered SS phase in terms of cavity amplitude Re[α] and Im[α] forms a circle. The profile exhibits a λ/2 period and corresponds to the positions changing continuously along the y axis by varying arg(α), demonstrating spontaneous continuous translational symmetry breaking. (c) Momentum-exchange and spin-momentum-mixing interactions between controlled spin atoms in different momentum states, including |k0⟩=|0,0⟩, |k1⟩=|±kL,kL⟩, and |k2⟩=|±kL,−kL⟩.

Fig. 2. (a) Cavity-mediated two dynamical SOC processes between the atomic zero-momentum |↑⟩ states and four nonzero-momentum |↓⟩ states. (b) Schematic representation of momentum space excitation paths with respect to two Raman channels (dashed and solid lines).

Fig. 3. (a) Phase diagram of the ground state on the g1-g2 parameter plane. The solid (dashed) line marks numerical (analytical) phase boundaries. (b) and (c) show, respectively, g1 dependence of |α| and N↓ for different values of g2.

Fig. 4. (a) g1 dependence of collective excitations ϵ1 (ϵ2) of lowest (middle) branches with g1/g2=1. (b) The lowest branch ϵ1 and (c) order parameter Θ as a function of g1 for different values of g2.

Fig. 5. The condensates wave functions of the PW (row 1) and SS phases (row 2) for (g1,g2)=(0, 7)EL/ℏ and (7, 7)EL/ℏ in spin-↓ atom, respectively. Columns 1–3 show the density ρ↓ (units of cm−2), relative phase Δϕ, and momentum distribution.