Emergent and broken symmetries of atomic self-organization arising from Gouy phase shifts in multimode cavity QED

Optical cavities can induce photon-mediated interactions among intracavity-trapped atoms. Multimode cavities provide the ability to tune the form of these interactions, e.g., by inducing a nonlocal sign-changing term to the interaction. By accounting for the Gouy phase shifts of the modes in a nearly degenerate, confocal, Fabry-Pérot cavity, we provide a theoretical description of this interaction, along with additional experimental confirmation to complement that presented in the companion paper [Y. Guo et al., Phys. Rev. Lett. 122, 193601 (2019)]. Furthermore, we show that this interaction should be written in terms of a complex order parameter, allowing for a $\mathrm{U}\left(1\right)$ symmetry to emerge. This symmetry corresponds to the phase of the atomic density wave arising from self-organization when the cavity is transversely pumped above a critical threshold power. We show theoretically and experimentally how this phase depends on the position of the Bose-Einstein condensate within the cavity and discuss mechanisms that break the $\mathrm{U}\left(1\right)$ symmetry and lock this phase. We then consider alternative Fabry-Pérot multimode cavity geometries (i.e., beyond the confocal) and schemes with more than one pump laser and show that these provide additional capabilities for tuning the cavity-meditated interaction among atoms, including the ability to restore the $\mathrm{U}\left(1\right)$ symmetry despite the presence of symmetry-breaking effects. These photon-mediated interactions may be exploited for realizing quantum liquid crystalline states and spin glasses using multimode optical cavities.

Figure 1

Magnitude of $|G^{+}({\mathbf{r}}_0,{\mathbf{r}}_0,2i{\theta }_0\left(z_0\right))|$ as a function of radius $r_0$ (vertical axis) and distance along the cavity $z$ (horizontal axis). Red dashed lines mark solutions to Eq. (28), indicating contours at which there is a degeneracy between the sine and cosine density-wave patterns.

Figure 2

Real (blue solid line) and imaginary (orange dotted line) parts of $G_{\mathrm{avg}}^{+}$ for a BEC at a transverse plane ($z_0/z_R=0.3$) located away from the cavity midplane and with a Gaussian width of ${\sigma }_A/w_0=0.1$. The black dashed line marks zero. Note that $\mathrm{Re}\left\{G_{\mathrm{avg}}^{+}\right\}$ and $\mathrm{Im}\left\{G_{\mathrm{avg}}^{+}\right\}$ do not vanish at the same point.

Figure 3

(a) Example of the experimental phase and amplitude of the cavity emission. The waist $w_0$ of the ${\mathrm{TEM}}_{00}$ in the image plane is indicated. Here $\mathrm{\Delta }\varphi =-2{\varphi }_A$ is computed from the difference in phase between the regions marked by the black circle and the average phase of the gray rectangular region. They correspond, respectively, to the local ${\mathcal{G}}^{+}(\mathbf{r},{\mathbf{r}}^{\prime },0)$ and nonlocal ${\mathcal{G}}^{+}(\mathbf{r},{\mathbf{r}}^{\prime },2i{\theta }_0)$ contributions to the light field in Eq. (21). (b) Color wheel for illustrating the electric field amplitude and phase. (c) Measured phase difference between the local and nonlocal contributions to the cavity field as a function of the distance $r$ of a single BEC from the cavity axis. The data shown here are taken with cavity detuning ${\mathrm{\Delta }}_{Q_0}=-120\mathrm{MHz}$. The distance is extracted by fitting to the amplitude of the measured field. The path of $\mathbf{r}$ taken in the $x-y$ plane from the cavity axis is shown as an black dashed arrow in (a).

Figure 4

(a) Reconstructed cavity superradiance from a BEC with length on the order of the cavity waist $w_0$. The phase of the atomic density wave varies smoothly from one side of a node to the other. Note that the gas is positioned to only one side of the cavity axis. (b) Line cut of the phase winding as indicated by white dashed line in (a). Here $r$ denotes the distance from the cavity axis.

Figure 5

Diagram illustrating evolution of cavity frequencies versus changing cavity length. Black vertical lines at the top of the panel indicate different longitudinal resonances, separated by one free spectral range. As one moves from the planar limit (top, $L\ll R$) to the concentric limit (bottom, $L=2R$), the different transverse modes within a given mode family split and cross those from higher longitudinal families. Some degenerate configurations are indicated by horizontal blue lines.