Cavity-Quantum-Electrodynamical Toolbox for Quantum Magnetism

The recent experimental observation of spinor self-ordering of ultracold atoms in optical resonators has set the stage for the exploration of emergent magnetic orders in quantum-gas-cavity systems. Based on this platform, we introduce a generic scheme for the implementation of long-range quantum spin Hamiltonians composed of various types of couplings, including Heisenberg and Dzyaloshinskii-Moriya interactions. Our model is composed of an effective two-component Bose-Einstein condensate, driven by two classical pump lasers and coupled to a single dynamic mode of a linear cavity in a double $\mathrm{\Lambda }$ scheme. Cavity photons mediate the long-range spin-spin interactions with spatially modulated coupling coefficients, where the latter ones can be tuned by modifying spatial profiles of the pump lasers. As experimentally relevant examples, we demonstrate that by properly choosing the spatial profiles of the pump lasers achiral domain-wall antiferromagnetic and chiral spin-spiral orders emerge beyond critical laser strengths. The transition between these two phases can be observed in a single experimental setup by tuning the reflectivity of a mirror. We also discuss extensions of our scheme for the implementation of other classes of spin Hamiltonians.

Figure 1

Schematic view of a transversely pumped multicomponent BEC in 2D inside a cavity. The inset depicts the internal atom-photon coupling scheme in a double $\mathrm{\Lambda }$ configuration.

Figure 2

The phase diagram of the system (a) and domain-wall AFM spin texture (b). (a) The phase diagram of the system is shown in the $\{R,\sqrt{N}{\eta }_0/{\omega }_r\}$ plane, where the color code indicates the rescaled cavity field amplitude $|{\alpha }_{\mathrm{ss}}|/\sqrt{N}$. (b) The projection of the normalized local spin $\tilde{\mathbf{s}}(\mathbf{r})$ on the ${\tilde{s}}_x\text{−}{\tilde{s}}_z$ plane is displayed as a function of $\mathbf{r}$ for two standing-wave pump lasers along the $y$ direction corresponding to $R=1$. The color code indicates the spin angle with respect to the ${\tilde{s}}_x$ axis, $\phi ={\tan }^{-1}({\tilde{s}}_z/{\tilde{s}}_x)$. Inside each domain, all spins are oriented with an angle ${\phi }_D\simeq 0.081\pi$ or $\pi -{\phi }_D$, while in boundaries they rotate quite rapidly. The parameters are set to $({\mathrm{\Delta }}_c,\kappa ,N{\mathcal{G}}_0^2/{\mathrm{\Delta }}_{1,2},\delta )=(-10,5,-1,-0.1){\omega }_r$ and $\sqrt{N}{\eta }_0=3.8{\omega }_r$, with the self-consistent field amplitude $|{\alpha }_{\mathrm{ss}}|/\sqrt{N}\simeq 0.22$. Here, $N$ is the total particle number and ${\omega }_r\equiv \hslash k_c^2/2M$.

Figure 3

Emergent SS texture. (a) The normalized local spin $\tilde{\mathbf{s}}(\mathbf{r})$ is shown for two counterpropagating pump lasers along the $y$ direction corresponding to $R=0$. The spin exhibits a transverse conical SS texture with magnetic domains. The latter is most clearly evident in the projections of $\tilde{\mathbf{s}}(\mathbf{r})$ in the ${\tilde{s}}_x\text{−}{\tilde{s}}_z$ and ${\tilde{s}}_y\text{−}{\tilde{s}}_z$ planes as shown in (b). The spin does a full $2\pi$ rotation in the ${\tilde{s}}_x\text{−}{\tilde{s}}_y$ plane along the $y$ direction over one ${\lambda }_c$, as can be seen in (c). The spin projection in the ${\tilde{s}}_x\text{−}{\tilde{s}}_y$ plane for $R=0.2$ is illustrated in (d) with pronounced magnetic domains. The color code in each figure indicates the respective spin angle, e.g., $\phi ={\tan }^{-1}({\tilde{s}}_y/{\tilde{s}}_x)$ in (c) and (d). The parameters are the same as Fig. 2, with the self-consistent field amplitude $|{\alpha }_{\mathrm{ss}}|/\sqrt{N}\simeq 0.14$ and 0.21 for $R=0$ and 0.2, respectively.