The experimental realization of ultracold atomic gases in optical cavities has unveiled unprecedented opportunities for simulating complex quantum matter [1–5], providing powerful solvers for revealing fundamental physics [6–8] and broad applications in quantum technology [9–11]. Among these advancements, the supersolid, which uniquely combines the dissipationless flow of superfluid and the long-range periodic density modulation of crystalline order, stands as one of the most enigmatic quantum states of matter [12]. The core challenge for supersolidity is requiring spontaneously breaking two mutually exclusive $U(1)$ symmetries, and its experimental exploration has undergone a tortuous development in solid helium [13–15], despite theoretical predictions dating back over half a century [16,17]. Recently, supersolidity has been extensively studied both theoretically and experimentally in frustrated quantum magnets [18], high-pressure deuterium [19], and synthetic materials [20–22].

The extraordinary capability of ultracold atoms, enriched with artificial gauge fields [23–28], dynamical spin–orbital coupling (SOC) [29–33], and intrinsic dipolar interaction [34–37], provides new opportunities for exploration of supersolid phases and advanced many-body physics [38–43]. In particular, the supersolid phase with a gapless Goldstone mode has been experimentally realized in spinless Bose–Einstein condensate (BEC) coupled to two noninterfering standing-wave optical cavities [44,45] and two modes of the ring cavity [46]. The mechanism for generating $U(1)$ symmetry employs two ${\mathcal{Z}}_2$ symmetries with strictly equal couplings for two-mode cavities, analogous to constructing the $U(1)$ symmetric $XY$ model from two ${\mathcal{Z}}_2$ symmetric Ising models [44,45]. Recently, promising routes for the realization of supersolid phases by utilizing a confocal multimode cavity [47–51] and axially elongated dipolar quantum droplets [52–55] have been extensively explored. The unambiguous supersolidity for experimental observation and the corresponding enigmatic fundamental quantum properties are yet to be fully explored, despite rapid theoretical proposals and experimental advances ranging from condensed-matter physics to ultracold quantum gases and cavity-QEDs.

In this paper, we propose a readily implementable experimental scheme to realize spin-momentum-mixing interactions with self-ordered supersolid square (SS) phase in spin-1/2 condensates coupled to an optical cavity. Unlike the experimentally observed Dicke superradiance with discrete ${\mathcal{Z}}_2$ symmetry breaking [56–58], the self-ordered superradiant phase transition is fully characterized by the two-component Tavis–Cummings model (TCM), indicative of broken continuous $U(1)$ symmetry. Notably, the laser configuration proposed here is akin to the pioneering experiment for realizing dynamical SOC with respect to the Dicke model [33]. Although supersolidity has been achieved for spinless BEC coupled to multimode cavities [44–46], which require strictly identical coupling strengths, our method necessitates only a simpler laser configuration and predicts the existence of supersolid phase in a large parameter regime.

Remarkably, our scheme introduces novel cavity-mediated spin-momentum-mixing interactions, characterized by strong spin-momentum correlations. In this process, pairs of zero-momentum atoms undergo spin flips into the same internal state while transitioning to distinct momentum modes. This process is analogous to, but distinct from collective momentum-exchange processes in atom–cavity systems [59] and weak spin-exchange collisions in spinor BEC [60]. Specifically, the two-mode squeezed vacuum state is realized for different momentum modes, with a large squeezing parameter. Our results significantly broaden the experimental scope for investigating multipartite entanglement between atoms with opposite momenta [61–65] and provide a unique platform for designing versatile quantum simulators within the realm of ultracold quantum gases [66–69].

We consider an $N$ bosonic atomic ${}^{87}\mathrm{Rb}$ BEC consisting of two ground and two excited states trapped in a high-finesse optical cavity. Figure 1(a) illustrates the level diagram of atoms. An applied bias magnetic field $\mathbf{B}$ breaks the degeneracy of the ground (excited) state manifold labeled as $|\uparrow ⟩$ and $|\downarrow ⟩$ ($|e_{\uparrow }⟩$ and $|e_{\downarrow }⟩$). The produced Zeeman shift between two hyperfine states $|\uparrow ⟩$ and $|\downarrow ⟩$ is given by $\hslash {\omega }_Z$ with their magnetic quantum numbers fulfilling $m_{\uparrow }=m_{\downarrow }+1$ and $m_{e_{\uparrow }}=m_{e_{\downarrow }}+1$. The transitions between $|\downarrow ⟩\leftrightarrow |e_{\uparrow }⟩$ and $|\uparrow ⟩\leftrightarrow |e_{\downarrow }⟩$ are driven by two transverse $\sigma$-polarized pump beams along the $y$ axis, with Rabi frequencies ${\mathrm{\Omega }}_{\mathrm{1,2}}(y)={\mathrm{\Omega }}_{\mathrm{1,2}}{e}^{i{k}_{L}y}$. Here, $k_L=2\pi /\lambda$ is the wave vector of the laser field with $\lambda$ being the wavelength. The laser frequency difference is set to $2{\omega }_Z$ for compensating Zeeman shift. Dynamical SOC is crafted through the coupling of $|\sigma ⟩\leftrightarrow |e_{\sigma }⟩$ ($\sigma =\uparrow$ and $\downarrow$) transitions by a $\pi$-polarized standing-wave cavity with the single atom–cavity coupling $g(x)=g\cos (k_{L}x)$. Collective Bragg scattering into the $\sigma$-polarized cavity mode is mitigated under the conditions $|{\omega }_Z/g|\gg 1$ [29].

For large atom–pump detuning $|\mathrm{\Delta }|\gg \{g,{\mathrm{\Omega }}_{\mathrm{1,2}}\}$, the atomic excited states $|e_{\uparrow ,\downarrow }⟩$ can be adiabatically eliminated and replaced by their steady-state solutions. Incorporating short-range collisional interactions, the effective many-body Hamiltonian of the atom–cavity reads $$\begin{gathered} {\widehat{\mathcal{H}}}_0=\hslash {\mathrm{\Delta }}_c{\widehat{a}}^{†}\widehat{a}+\sum _{\sigma {\sigma }^{\prime }}\int \mathrm{d}\mathbf{r}{\widehat{\psi }}_{\sigma }^{†}(\mathbf{r})[{\widehat{h}}_{\sigma {\sigma }^{\prime }}+V_b(\mathbf{r}){\delta }_{\sigma {\sigma }^{\prime }}]{\widehat{\psi }}_{{\sigma }^{\prime }}(\mathbf{r}) \\ +\frac{1}{2}\sum _{\sigma {\sigma }^{\prime }}\frac{4\pi {\hslash }^2{a}_{\sigma {\sigma }^{\prime }}}{m}\int \mathrm{d}\mathbf{r}{\widehat{\psi }}_{\sigma }^{†}(\mathbf{r}){\widehat{\psi }}_{{\sigma }^{\prime }}^{†}(\mathbf{r}){\widehat{\psi }}_{{\sigma }^{\prime }}(\mathbf{r}){\widehat{\psi }}_{\sigma }(\mathbf{r}), \end{gathered}$$(1)where $m$ is the mass of the atom, $\widehat{a}$ is the annihilation operator of the cavity, and ${\widehat{\psi }}_{\sigma }$ represents the annihilation bosonic field operator for the spin-$\sigma$ atom. ${\mathrm{\Delta }}_c$ is the pump–cavity detuning, $V_b(\mathbf{r})$ is the external uniform potential of the box trap, and $a_{\sigma {\sigma }^{\prime }}$ are $\mathrm{s}$-wave scattering lengths for intraspecies ($\sigma ={\sigma }^{\prime }$) and interspecies ($\sigma \ne {\sigma }^{\prime }$) spin atoms. Additionally, the single-particle Hamiltonian satisfies $$\widehat{h}=\frac{{\mathbf{p}}^2}{2\mathit{m}}\widehat{I}+\hslash \left(\begin{array}{cc}{\widehat{M}}_0(x)+\delta /2& {\widehat{M}}_{-}(x,y)\\ {\widehat{M}}_{-}^{†}(x,y)& {\widehat{M}}_0(x)-\delta /2\end{array}\right),$$where $\delta$ is the tunable two-photon detuning, ${\widehat{M}}_{-}(x,y)=(g_1\widehat{a}{e}^{i{k}_{L}y}+g_2{\widehat{a}}^{†}{e}^{-i{k}_{L}y})\cos (k_{L}x)$ is cavity-mediated Raman coupling with $g_{\mathrm{1,2}}=-g{\mathrm{\Omega }}_{\mathrm{1,2}}/\mathrm{\Delta }$ being the maximum scattering rate, and ${\widehat{M}}_0(x)=-U_0{\cos }^2(k_{L}x){\widehat{a}}^{†}\widehat{a}$ is the optical lattice with Stark shift $U_0=g^2/\mathrm{\Delta }$. The spin-flip interaction ${\widehat{M}}_{-}$ integrates two dynamical SOCs [29,30], emerging from the interference between the cavity and two classical pump fields. This interplay of dual dynamical SOC fosters spatially translational invariant crystalline orders within the condensates wave functions.

To advance, cavity field $\widehat{a}$ is adiabatically eliminated in the far dispersive limit $|{\mathrm{\Delta }}_c/g_{\mathrm{1,2}}|\gg 1$ since its dynamical evolution is much faster than external atomic motion. The steady-state intracavity amplitude satisfies $\alpha =⟨\widehat{a}⟩=(g_1{\mathrm{\Xi }}_1^{†}+g_2{\mathrm{\Xi }}_2)/(-{\tilde{\mathrm{\Delta }}}_c+i\kappa )$. Here, $\kappa$ is the cavity decay, ${\tilde{\mathrm{\Delta }}}_c={\mathrm{\Delta }}_c-U_0ℬ$ represents the $N$-dependent dispersive shift of the cavity with $ℬ={\sum }_{\sigma }⟨{\psi }_{\sigma }|{\cos }^2(k_{L}x)|{\psi }_{\sigma }⟩$, and ${\mathrm{\Xi }}_1=⟨{\psi }_{\uparrow }|\cos (k_{L}x)e^{i{k}_{L}y}|{\psi }_{\downarrow }⟩/N$ and ${\mathrm{\Xi }}_2=⟨{\psi }_{\uparrow }|\cos (k_{L}x)e^{-i{k}_{L}y}|{\psi }_{\downarrow }⟩/N$ are order parameters dictating the configurations of ground-state condensates wave functions. Consequently, the self-ordered superradiance corresponds to a finite $\alpha$ due to ${\mathrm{\Xi }}_{\mathrm{1,2}}\ne 0$ for plane wave (PW) and SS phases. We neglect the atom–cavity entanglement for the steady-state solution considering moderate photon emissions with $|\alpha |\sim 1$. The entanglement between the photon and condensates, along with the quantum noise of the cavity can be ignored when $|{\tilde{\mathrm{\Delta }}}_c/\kappa |\gg 1$ [70,71].

For a deeper understanding of underlying physics, the self-organized superradiance can be comprehensively characterized by the two-component TCM Hamiltonian ($\hslash =1$): $${\widehat{\mathcal{H}}}_1={\tilde{\mathrm{\Delta }}}_c{\widehat{a}}^{†}\widehat{a}+{\omega }_0{\widehat{J}}_z+\frac{g_1}{\sqrt{2}}(\widehat{a}{\widehat{J}}_{-}^{(1)}+\frac{g_2}{g_1}{\widehat{a}}^{†}{\widehat{J}}_{-}^{(2)}+\text{H.c.}),$$(2)which describes two-component two-level bosonic atoms coupled to a quantized cavity. ${\omega }_0=2E_L/\hslash -\delta$ is the effective detuning of the atomic field, and ${\widehat{J}}_{-}^{(1)}={\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\downarrow ,1}$, ${\widehat{J}}_{-}^{(2)}={\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\downarrow ,2}$, and ${\widehat{J}}_z=({\widehat{b}}_{\downarrow ,1}^{†}{\widehat{b}}_{\downarrow ,1}+{\widehat{b}}_{\downarrow ,2}^{†}{\widehat{b}}_{\downarrow ,2}-{\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\uparrow ,0})/2$ are collective spin operators. Here, ${\widehat{b}}_{\uparrow ,0}$ and ${\widehat{b}}_{\downarrow ,1}$ (${\widehat{b}}_{\downarrow ,2}$) represent the annihilation bosonic operators for zero- and nonzero-momentum with $|k_x,k_y⟩=|\pm k_L,k_L⟩$ ($|\pm k_L,-k_L⟩$) in $|\uparrow ⟩$ and $|\downarrow ⟩$ atoms. Accordingly, the analytically critical Raman coupling for the two-component TCM satisfies ${[g_1^2+g_2^2]}_{\mathrm{cr}}=2{({\tilde{\mathrm{\Delta }}}^2+{\kappa }^2)}^{1/2}{\omega }_0/N$. Interestingly, the threshold of superradiant phase transition is a combination of two cavity-mediated Raman couplings, revealing a high ground-state degeneracy.

Indeed, ${\widehat{\mathcal{H}}}_1$ is realized within the single recoil scattering limit [56–58], excluding contact interactions as well. The microscopic picture of light scattering in superradiance coherently transfers the atomic motional ground state $|\mathrm{0,0}⟩$ in $|\uparrow ⟩$ to the equal-energy excited momentum states $|\pm k_L,\pm k_L⟩$ in $|\downarrow ⟩$ state, induced by the cavity-mediated dynamical SOC mechanism, as illustrated in Fig. 2(a). In contrast to the Dicke mode, which allows two excitation paths from the momentum states $|\mathrm{0,0}⟩$ to $|\pm k_L,\pm k_L⟩$ simultaneously [56], the nonzero momentum states $|\pm k_L,k_L⟩$ and $|\pm k_L,-k_L⟩$ are separately generated by the two cavity-mediated Raman processes with corresponding operators ${\widehat{a}}^{†}{\widehat{J}}_{+}^{(1)}$ and ${\widehat{a}}^{†}{\widehat{J}}_{+}^{(2)}$ [Fig. 2(b)].

Compared to Dicke superradiance, which hosts the gapped roton mode due to the discrete ${\mathcal{Z}}_2$ symmetry breaking [57], the $U(1)$-symmetric Hamiltonian ${\widehat{\mathcal{H}}}_1$ is gauge invariant under the unitary transformation ${\mathcal{R}}_{\theta }^{†}(\widehat{a},{\widehat{J}}_{-}^{(1)},{\widehat{J}}_{-}^{(2)}){\mathcal{R}}_{\theta }=(\widehat{a}{e}^{i\theta },{\widehat{J}}_{-}^{(1)}{e}^{-i\theta },{\widehat{J}}_{-}^{(2)}{e}^{i\theta })$ with the transformation operator ${\mathcal{R}}_{\theta }=\exp [i\theta ({\widehat{a}}^{†}\widehat{a}-{\widehat{b}}_{\downarrow ,1}^{†}{\widehat{b}}_{\downarrow ,1}+{\widehat{b}}_{\downarrow ,2}^{†}{\widehat{b}}_{\downarrow ,2})]$ [72,73]. A key feature of the two-component TCM is that it inherently possesses $U(1)$ symmetry, which does not necessitate two strictly equal Raman couplings ($g_1=g_2$). This proposal, achievable with current experimental setups for studying dynamical SOC for Dicke superradiance utilizing identical laser configurations [33], contrasts with the realization of checkerboard supersolidity, which hinges on constructing a $U(1)$ symmetry via two ${\mathcal{Z}}_2$ symmetries with precisely matched atom–cavity couplings in multimode resonators [44,45]. Notably, the analogous construction of $U(1)$ symmetry in light–matter systems, corresponding to rich quantum multicritical phenomena, has been both theoretically studied and experimentally realized in recent works [74,75].

To characterize new properties of superradiance, one can derive cavity-mediated collective interactions of atomic fields by integrating out the cavity field: $${\widehat{\mathcal{H}}}_2=\frac{1}{2}\int \mathrm{d}\mathbf{r}\mathrm{d}{\mathbf{r}}^{\prime }{U}_{\mathrm{eff}}(\mathbf{r},{\mathbf{r}}^{\prime })\{V_1{\widehat{S}}_{-}(\mathbf{r}){\widehat{S}}_{+}({\mathbf{r}}^{\prime })+V_2{\widehat{S}}_{+}(\mathbf{r}){\widehat{S}}_{-}({\mathbf{r}}^{\prime })+V_3[{\widehat{S}}_{+}(\mathbf{r}){\widehat{S}}_{+}({\mathbf{r}}^{\prime })+\text{H.c.}]\},$$(3)where $U_{\mathrm{eff}}(\mathbf{r},{\mathbf{r}}^{\prime })=2\cos (k_{L}x)\cos (k_L{x}^{\prime })e^{i{k}_L(y-y^{\prime })}$ is the long-range potential, ${\widehat{S}}_{+}(\mathbf{r})={\widehat{\psi }}_{\uparrow }^{†}(\mathbf{r}){\widehat{\psi }}_{\downarrow }(\mathbf{r})$ is the spin operator, and $V_{j=\mathrm{1,2}}=-{\tilde{\mathrm{\Delta }}}_c{g}_j^2/({\tilde{\mathrm{\Delta }}}_c^2+{\kappa }^2)$ and $V_3=-{\tilde{\mathrm{\Delta }}}_c{g}_1{g}_2/({\tilde{\mathrm{\Delta }}}_c^2+{\kappa }^2)$ are tunable strengths of two-body interactions. Remarkably, the first two terms of ${\widehat{\mathcal{H}}}_2$ represent the long-range spin-exchange interaction between spin-$\uparrow$ and $\downarrow$ atoms, which has been extensively studied including atom–cavity superradiance [76,77]. The third term introduces a two-axis twisting interaction via a superradiant photon-exchange process originally proposed by Kitagawa and Ueda [78], which does not conserve the atom number in the individual spin state [78–80]. It is worth noting that these interactions provide a new avenue for exploring self-organized superradiance in spinor condensates.

By performing a few-mode expansion, the spin-momentum-mixing interactions in terms of spin and momentum modes are given by $${\widehat{\mathcal{H}}}_3=\sum _{j=\mathrm{1,2}}\frac{V_j}{2}{\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\downarrow ,j}^{†}{\widehat{b}}_{\uparrow ,0}{\widehat{b}}_{\downarrow ,j}+\frac{V_3}{2}({\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\downarrow ,1}{\widehat{b}}_{\downarrow ,2}+\text{H.c.}).$$(4)The first term denotes collective momentum-exchange interactions [Fig. 1(c)] as experimentally realized in Ref. [59], commonly referred to as one-axis twisting. Importantly, the second term originating from two-axis twisting introduces a novel mechanism for the deterministic generation of entangled momentum-correlated pairs from zero-momentum condensates. These controllable spin-momentum-mixing interactions are analogous to, yet distinct from, spin-mixing interaction in spin-1 BEC under the single-mode approximation [60].

We note that Hamiltonian Eq. (5) dominates self-organized condensates wave functions of ground states subsequent to superradiance, significantly surpassing the typical two-body collisional interaction. In particular, the supersolid SS phase with spatially translationally invariant crystalline order disappears when the two-axis twisting interaction $V_3=0$. By tuning the ratio of the cavity-mediated Raman coupling $g_1/g_2$, the strength of $V_3$ can be gradually increased from zero to a maximum value of $\max (|V_1|,|V_2|)$. Remarkably, the magnitude of $N{V}_3$ is typically of the order of tens of kilohertz, vastly exceeding the spin-mixing rate (of the order of tens of hertz) achievable in experimental engineering twin-Fock states for ${}^{87}\mathrm{Rb}$ condensates [81]. Compared to the experimentally realized momentum-exchange interactions [59], Eq. (5) undergoing cavity-mediated spin-momentum-mixing dynamics is deeply entangled, enabling the creation of novel nonclassical states highly correlated in different momentum modes, e.g., spatially separated deterministic entanglement [61–65].

To proceed further, the two-axis twisting interaction can be interpreted as the parametric conversion process for generating entangled pairs between $|\pm k_L,k_L⟩$ and $|\pm k_L,-k_L⟩$ momentum modes. This corresponds to the coherent spin-momentum-changing collision process, $2|\mathrm{0,0}⟩\leftrightarrow |\pm k_L,k_L⟩+|\pm k_L,-k_L⟩$. Under the undepleted pump approximation, the initially highly populated $|\uparrow ⟩$ state acts as a classical source, which is modeled by replacing ${\widehat{b}}_{\uparrow ,0}\to \sqrt{N}$ in ${\widehat{\mathcal{H}}}_3$. The effective Hamiltonian describing the parametric conversion process is given by $${\widehat{\mathcal{H}}}_4={\omega }_1{\widehat{b}}_{\downarrow ,1}^{†}{\widehat{b}}_{\downarrow ,1}+{\omega }_2{\widehat{b}}_{\downarrow ,2}^{†}{\widehat{b}}_{\downarrow ,2}+\frac{N{V}_3}{2}({\widehat{b}}_{\downarrow ,1}{\widehat{b}}_{\downarrow ,2}+\text{H.c.}),$$(5)with ${\omega }_i={\omega }_0+N{V}_i/2$. In the Fock state representation, the two-mode squeezed vacuum state emerging from the initial vacuum state $|\mathrm{0,0}⟩$ is $$\begin{gathered} |{\psi }_S⟩=\exp [r({\widehat{b}}_{\downarrow ,1}{\widehat{b}}_{\downarrow ,1}-{\widehat{b}}_{\downarrow ,1}^{†}{\widehat{b}}_{\downarrow ,1}^{†})]|\mathrm{0,0}⟩ \\ =\frac{1}{\cosh r}\sum _{n=0}^{\infty }{(-\tanh r)}^n|n,n⟩, \end{gathered}$$(6)where the squeezing parameter $r$ satisfies $\tanh r=[\sqrt{{({\omega }_1+{\omega }_2)}^2-N^2{V}_3^2}-({\omega }_1+{\omega }_2)]/(N{V}_3)$. This state represents a coherent superposition of strictly particle number correlated Fock states across different momentum modes and exhibits thermal photon statistics in both the $|\pm k_L,k_L⟩$ and $|\pm k_L,-k_L⟩$ modes. We should note that the squeezing parameter $r$ increases rapidly from zero to a large value as $V_3$ grows, which may enable the study of Heisenberg-limit scaling ($1/N$) using two-mode squeezing for enhanced metrology.

To gain more insight, Hamiltonian ${\widehat{\mathcal{H}}}_3$ can be simplified to a one-axis twisting model in well-defined orbital angular momentum operators ${\widehat{\mathcal{H}}}_{\mathrm{eff}}=V_3{\widehat{L}}_{+}{\widehat{L}}_{-}/4=V_3({\widehat{L}}^2-{\widehat{L}}_z^2+{\widehat{L}}_z)/4$ when $g_1=g_2$. The raising and lowering operators ${\widehat{L}}_{+}={\widehat{L}}_{-}^{†}=\sqrt{2}({\widehat{b}}_{\downarrow ,2}^{†}{\widehat{b}}_{\uparrow ,0}+{\widehat{b}}_{\uparrow ,0}^{†}{\widehat{b}}_{\downarrow ,1})$ obey the angular momentum commutation relations $[{\widehat{L}}_{+},{\widehat{L}}_{-}]=2{\widehat{L}}_z$ and $[{\widehat{L}}_z,{\widehat{L}}_{\pm }]=\pm {\widehat{L}}_{\pm }$ with ${\widehat{L}}_z={\widehat{b}}_{\downarrow ,1}^{†}{\widehat{b}}_{\downarrow ,1}-{\widehat{b}}_{\downarrow ,2}^{†}{\widehat{b}}_{\downarrow ,2}$. We should emphasize that the operator $\widehat{L}$ acts on the coupled spin-1 spin-momentum degree of freedom in contrast to the spinor condensates.

For experimental observation, the transition into superradiant phases, characterized by a nonzero photon number, can be monitored by measuring the inherent leakage of the cavity [56]. Distinguishing between PW and SS phases can be achieved by detecting atomic momentum distribution via spin-sensitive absorption images [58] or using Bragg scattering techniques [28]. Additionally, the presence of the zero-energy Goldstone mode insensitive to cavity dissipation can be identified through spectroscopic measurement for ultracold quantum gases [45]. Notably, the SS phase exhibiting the spatial periodicity of $\lambda /2$ is different from the previously realized $\lambda$ periodic ${\mathcal{Z}}_2$-broken checkerboard lattice supersolid involving both scalar and spinor condensates within a cavity [56–58].

We study ground-state phases by solving Gross–Pitaevskii equations via imaginary time evolution. In a mean-field framework, atomic and cavity field operators are replaced by ${\psi }_{\sigma }=⟨{\psi }_{\sigma }⟩$ and $\alpha =⟨\widehat{a}⟩$ and calculated in a self-consistent manner. We consider $N{=10}^5$ BEC initially in the $|\uparrow ⟩$ state and confined in a circular optical box trap [82,83]. Unlike the pancake-shaped harmonic trap, the elaborately selected uniform potential preserves a continuous translational symmetry. Specifically, we take single-photon recoil energy $E_L/\hslash =3.53\mathrm{kHz}(2\pi )$ with the wavelength $\lambda =2\pi /k_L=803.2\mathrm{nm}$, cavity decay rate $\kappa =100E_L/\hslash$, two-photon detuning $\delta =-2E_L/\hslash$, Stark shift $U_0=10E_L/\hslash$, and pump–cavity detuning ${\tilde{\mathrm{\Delta }}}_c=U_{0}N/2$. The $\mathrm{s}$-wave scattering lengths for collisional interactions are $a_{\downarrow \downarrow }=a_{\uparrow \downarrow }\approx a_{\uparrow \uparrow }=50a_B$ with $a_B$ being the Bohr radius. To this end, the free controllable parameters are Raman couplings $g_1$ and $g_2$.

Figure 3(a) summarizes the ground-state phase diagram. The interplay of Raman couplings $g_1$ and $g_2$ leads to two self-ordered superradiant PW and SS phases with nonzero intracavity amplitude $\alpha \ne 0$. In the absence of superradiance, the superfluid phase is denoted by “N.” It is apparent that SS and PW phases appear symmetrically in the phase diagram, highlighting that the SS phase does not require $g_1/g_2=1$. This observation marks a departure from checkerboard supersolidity, which demands strictly equal couplings to maintain $U(1)$ symmetry in dual-mode cavities [44,45]. For a relatively small $g_1$ ($g_2$), the system enters the PW phase without a crystalline order induced by dynamical SOC [32,33].

Compared to the two-component TCM Hamiltonian Eq. (2), the ground-state structures of the cavity-condensate system are calculated from the full Hamiltonian, including the atomic contact interaction. We check that the density profiles for both PW and SS phases are almost unaffected by two-body collisional atom–atom interaction, owing to the dominance of cavity-mediated collective interactions. Indeed, the short-range collisional interaction is effectively decoupled from the cavity dynamics in the single-recoil scattering approximation [56–58]. Furthermore, our numerical simulations show that the threshold for superradiance is robust against the slight variations of $\mathrm{s}$-wave scattering length.

The transition from N to the self-ordered SS phase is attributed to superradiance in the two-component TCM, corresponding to the effective potential of ground states extending from a minimum in the origin to a sombrero shape with a circular valley of degenerate minima. The observed quarter circle on $g_1\text{-}{g}_2$ plotted in Fig. 3(a) further reveals the high ground-state degeneracy. The analytical threshold for superradiance (red dashed line) is in strong agreement with numerical simulations (black solid line), incorporating atomic collision and cavity dispersion. In Figs. 3(b) and 3(c), we show cavity amplitude ($|\alpha |$) and spin-$\downarrow$ population ($N_{\downarrow }$) versus $g_1$ for various $g_2$ values, serving as order parameters to distinguish between N and superradiant phases. This marks the breaking of $U(1)$ symmetry from vacuum to a finite value spontaneously. Additionally, a large $g_2$ lowers the $g_1$ threshold for superradiance, as predicted by analytical results of the two-component TCM.

To demonstrate the rigidity of self-ordered superradiant phases, we calculate the collective excitation of the two-component TCM. By diagonalizing the Hopfield–Bogoliubov matrix, we find that the system always exhibits three nonnegative eigenvalues corresponding to the lowest, middle, and highest branches of collective excitations. A gapless Goldstone mode of lowest (${ϵ}_1$) and gapped amplitude modes of the middle (${ϵ}_2$) branches are identified, arising from the spontaneous breaking of $U(1)$ symmetry in superradiance, as displayed in Fig. 4(a). Despite the inclusion of nonzero cavity dissipation, the lifetime of this zero-energy mode significantly exceeds that of the quantized cavity and is roughly undamped with $\mathrm{Im}({ϵ}_1)/\kappa \sim {10}^{-4}$. More importantly, this zero-energy Goldstone mode persists in superradiant phases regardless of whether $g_1$ equals $g_2$ or not [Fig. 4(b)], which greatly enhances the experimental feasibility in atom–cavity QEDs [44–46].

The order parameter of $\alpha$ characterizes superradiant phase transition from vacuum ($\alpha =0$) to a finite value ($\alpha \ne 0$), but it cannot discern the transition from the PW to the SS phase. To address this, we introduce an order parameter $\mathrm{\Theta }\equiv ⟨{\psi }_{\downarrow }|\cos (2k_{L}x)\cos (2k_{L}y)|{\psi }_{\downarrow }⟩/N$, which quantifies the periodic density modulation via the configuration of the condensates. In Fig. 4(c), we depict $g_1$ dependence of $\mathrm{\Theta }$ for different values of $g_2$. Compared to the PW phase with $\mathrm{\Theta }\sim 0$, the SS phase shows a large value of $\mathrm{\Theta }$, revealing a strong periodic density modulation of the ground state. Particularly for $g_1/g_2\ne 1$, the emergence of a self-ordered crystalline structure with respect to a zero-energy Goldstone mode is also confirmed. The large $g_2$ corresponds to a heightened $\mathrm{\Theta }$, signifying a strong self-ordered crystalline arrangement. Moreover, $\mathrm{\Theta }$ exhibits a sharp increase from N to SS phases, facilitating the natural monitoring of the superradiant phase transition.

Figure 5 shows typical condensates wave functions for two superradiant phases. We find that the self-ordered crystalline order always appears in the less populated spin-$\downarrow$ state since the $\uparrow$-component is populated dominantly in numerical simulations. The PW phase presents a nearly structureless density profile with staggered $\lambda$-periodic phase modulation along the $y$ axis and a $\lambda /2$-periodic density stripe along the $x$ axis, created by the cavity-induced optical lattice. For minimizing energy, the relative phase of condensates wave functions requires $\mathrm{\Delta }\varphi =\arg ({\psi }_{\downarrow })-\arg ({\psi }_{\uparrow })=k_{L}y+\pi$ ($k_{L}y$) when $\cos (k_{L}x)=1$ ($-1$) [29]. With this correspondence, the momentum-space distribution exhibits two peaks at $|\pm k_L,-k_L⟩$, as shown in Fig. 5(c). Furthermore, the weak 2D periodic density modulation is in good agreement with $\mathrm{\Theta }\sim 0$. Similar to TCM superradiance [32], the observed PW phase resulting from dynamical SOC is dominated by individual $g_2$ ($g_1$) Raman coupling with $g_2/g_1\gg 1$ ($g_1/g_2\gg 1$) in Fig. 3(a). Compared to the experimentally observed PW phase with dynamical SOC and a gapped roton mode [33], our finding demonstrates the spontaneous $U(1)$ symmetry breaking and gapless Goldstone mode excitation [Fig. 4(b)].

When $g_1$ and $g_2$ are comparable, the system transitions into an interesting SS phase characterized by significant periodic density modulation with a high $\mathrm{\Theta }$ value, distinguishing it from the PW phase. Notably, the self-ordered density profile depends on the cavity phase angle $\arg (\alpha )$, corresponding to the positions changing continuously along the $y$ axis [Fig. 1(b)]. This provides an unambiguous demonstration of supersolidity combined with rigidity of the gapless Goldstone mode. The phenomenon of self-organization in superfluid quantum gases that cavity amplitude $\alpha$ is self-consistently determined by the atomic wave function [56], originates from the broken continuous translational symmetry inherent in two-component TCM superradiance. For $\arg (\alpha )=0$, the sites of peak density satisfy $\cos (k_{L}y)\cos (k_{L}x)=+1(-1)$ with a corresponding relative phase $\mathrm{\Delta }\varphi =\pi (0)$. This setup yields atomic momentum distribution at $(k_x,k_y)=(\pm k_L,\pm k_L)$ and matches the configurations of the crystalline square lattice pattern. Although the momentum distribution mirrors that of the lattice checkerboard supersolid, which breaks the discrete ${\mathcal{Z}}_2$ translational symmetry, the self-ordered supersolid square phase in our system is characterized by the two-component TCM, corresponding to spontaneous breaking of continuous translational symmetry.

We present an experimental scheme to engineer strong spin-momentum-mixing interactions in cavity-coupled spinor condensates, intricately correlating spin and momentum modes via the superradiant photon-exchange process. In particular, we realize the self-ordered supersolid square phase with a combination of continuous translational symmetry breaking and the undamped gapless Goldstone mode, as described by two-component TCM superradiance. Our approach also highlights the enhanced experimental feasibility compared to pioneering studies and can be readily tested with existing experimental setups. In contrast to the realized dynamical SOC with discrete ${\mathcal{Z}}_2$ symmetry in Ref. [33], our work preserves continuous translational symmetry using the same laser configuration. Remarkably, the mechanism for generating supersolidity does not require equal couplings for the two cavity-mediated Raman processes, significantly simplifying experimental feasibility in cavity QEDs [44–51]. Moreover, we realize the two-mode squeezed vacuum state with the tunable squeezing parameter for different momentum modes. Compared to collective momentum-exchange interactions [59] or weak spin-mixing processes in spin-1 condensates [81], the emergence of cavity-mediated spin-momentum-mixing interactions opens new avenues for exploring entanglement-enhanced metrology [81], spin-momentum squeezing [84,85], and spatially separated multipartite entanglement using highly correlated momentum modes [61–65].