Spinor Self-Ordering of a Quantum Gas in a Cavity

We observe the joint spin-spatial (spinor) self-organization of a two-component Bose-Einstein condensate (BEC) strongly coupled to an optical cavity. This unusual nonequilibrium Hepp-Lieb-Dicke phase transition is driven by an off-resonant Raman transition formed from a classical pump field and the emergent quantum dynamical cavity field. This mediates a spinor-spinor interaction that, above a critical strength, simultaneously organizes opposite spinor states of the BEC on opposite checkerboard configurations of an emergent 2D lattice. The resulting spinor density-wave polariton condensate is observed by directly detecting the atomic spin and momentum state and by holographically reconstructing the phase of the emitted cavity field. The latter provides a direct measure of the spin state, and a spin-spatial domain wall is observed. The photon-mediated spin interactions demonstrated here may be engineered to create dynamical gauge fields and quantum spin glasses.

Figure 1

(a) Experimental setup and detection techniques. The two Raman pump beams (red and blue), polarized along the cavity axis, are combined and retroreflected off the same mirror to create a phase-stable lattice (purple). The cavity mode (green), imaged onto an EMCCD camera, interferes with a local oscillator at an angle (also green). This provides the spatial heterodyne signal (blue lines) for the holographic reconstruction of the cavity field amplitude and phase. Momentum of the BEC (scarlet) is absorption imaged in time of flight (scarlet beam). (b) Double Raman scheme for coupling two ${}^{87}\mathrm{Rb}$ Zeeman states. (c) Real-space illustration of the transition from randomly positioned atoms below threshold (left) to a checkerboard spinor order in an emergent 2D optical lattice above threshold (right-hand panels). Atoms are in a $\widehat{z}$ ($\widehat{x}$) spin-polarization state below (above) threshold, where $|\leftrightarrows ⟩=|\downarrow ⟩\pm |\uparrow ⟩$. The ${\mathbb{Z}}_2$-order breaking selects one of two states in which $|\to ⟩$ are at the sites colored black (right, top) or white (right, bottom). Dashed (solid) lines in left-hand panel are the nodes of the emergent cavity (pump) field. Solid lines in the right-hand panels are the nodes of the above-threshold 2D optical lattice. (d) Momentum-space illustration of spin state after sequential photon recoils from the pump and cavity fields. The $|\uparrow ⟩$ spin component has phase $\pm 1$ depending on which ${\mathbb{Z}}_2$ state emerges above threshold. Arrow colors depict pathway of optical transition in (b).

Figure 2

(a) Cavity emission detected by single photon counters versus time plotted with the concomitant linear increase in lattice depth (proportional to pump intensity). The superradiant transition threshold is at $t\approx 0.55\mathrm{ms}$. (b),(c) Spin-sensitive absorption images of the atomic cloud in time of flight reveal the optical density (OD) of the momentum distribution of both spin states at the times indicated in (a). (b) All atoms are in $|\downarrow ⟩$ below threshold and either at zero momentum or at $\mathbf{k}=(\pm 2k_r,0)$ due to pump-lattice diffraction. (c) Above threshold, atoms have undergone a spin flip to $|\uparrow ⟩$ accompanied by a $\mathbf{k}=\{(\pm k_r,\pm k_r);(\pm k_r,\mp k_r)\}$ momentum kick. The resulting Bragg peaks are spin colored in the same pattern as in Fig. 1.

Figure 3

Fringe amplitude factor $\chi$ as function of local oscillator frequency detuning ${\delta }_{\mathrm{LO}}$. The cavity is pumped above threshold at a detuning ${\mathrm{\Delta }}_c=-4\mathrm{MHz}$ from the ${\mathrm{TEM}}_{0,0}$ cavity resonance. The camera integration time is 2 ms. Insets show the spatial heterodyne signal—with local oscillator field subtracted for clarity—for both a maximal $\chi$ and where fringes average out at ${\delta }_{\mathrm{LO}}=3\mathrm{kHz}$. Error bars represent 1 standard deviation over five repetitions.

Figure 4

(a) Holographic reconstruction of cavity field amplitude and phase for a cavity locked near the ${\mathrm{TEM}}_{1,0}$ mode whose spatial profile $\mathrm{\Xi }(x,z)$ exhibits a sign flip at $x=0$. The phase of the right-hand lobe is defined as 0 with respect to the local oscillator. The phase shows a jump of exactly $\pi$ across the cavity center, demonstrating the fixed relative phase difference between the $\mathrm{\Theta }=\pm 1$ states with respect to the local oscillator phase. (b) Observed spin-density structure factor. The small-$k$ transverse-mode structure appears as a node in the first-order Bragg peaks. The combination of atomic and photonic observations indicates the existence of a domain wall in the spinor.