Dynamical Spin-Orbit Coupling of a Quantum Gas

We realize the dynamical 1D spin-orbit coupling (SOC) of a Bose-Einstein condensate confined within an optical cavity. The SOC emerges through spin-correlated momentum impulses delivered to the atoms via Raman transitions. These are effected by classical pump fields acting in concert with the quantum dynamical cavity field. Above a critical pump power, the Raman coupling emerges as the atoms superradiantly populate the cavity mode with photons. Concomitantly, these photons cause a backaction onto the atoms, forcing them to order their spin-spatial state. This SOC-inducing superradiant Dicke phase transition results in a spinor-helix polariton condensate. We observe emergent SOC through spin-resolved atomic momentum imaging and temporal heterodyne measurement of the cavity-field emission. Dynamical SOC in quantum gas cavity QED, and the extension to dynamical gauge fields, may enable the creation of Meissner-like effects, topological superfluids, and exotic quantum Hall states in coupled light-matter systems.

Figure 1

(a) Schematic of the experiment. Two Raman pump beams (red and blue arrows), polarized along the cavity axis, counterpropagate through a BEC of Rb (purple) inside a ${\mathrm{TEM}}_{00}$ cavity. The cavity emission (green arrow) is detected by a single-photon counter, and the atoms are imaged in time of flight by a CCD camera (not shown). (b) Level diagram illustrating the cavity-assisted Raman coupling between two hyperfine levels of ${}^{87}\mathrm{Rb}$ acting as the spin states. The counterpropagating running-wave nature of the pumps is explicitly notated by $e^{\pm ikx}$. See text for definitions of all quantities.

Figure 2

(a),(b) Momentum-space depiction of the emergence of SOC. (a) Initially atoms are in a spin-polarized state $|\downarrow ⟩$. (b) If the transverse pumping strength is sufficiently strong, SOC emerges and the spin components are in different momentum states. The $\pm$ sign of the $|\uparrow ⟩$ spin component indicates the ${\mathbb{Z}}_2$ symmetry-broken phase freedom. (c),(d) Energy-momentum dispersion relation of each spin state, transitioning from free (c) to coupled (d) dispersion bands. The coupling strength ${\widehat{\mathrm{\Omega }}}_{\mathrm{SOC}}$ is proportional to $\widehat{a}$ and ${\widehat{a}}^{†}$ and therefore arises dynamically as the atoms scatter pump photons into the cavity. The zero of the momentum has been shifted with respect to the lab frame by $-k_r/2$ in the plot; cf. the unitary transformations in Ref. [45].

Figure 3

(a) Cavity emission detected by single-photon counters (solid blue line) and optical power in the Raman beams (dashed black line), both as a function of time. Steady-state SOC persists up to a few milliseconds; oscillations are due to system dynamics as described in Ref. [49]. (b),(c) Spin-resolved momentum distribution in time of flight, taken at the points labeled in (a). All atoms are in $|\downarrow ⟩$ just below threshold (b). Above threshold (c), spin-up atoms have acquired a net momentum in the $x$ direction, as shown by the spin-colored Bragg peaks at nonzero momentum. Also shown are second-order diffraction peaks along the cavity direction due to the reverse Raman process.

Figure 4

(a) Cavity emission amplitude (solid blue line) and transverse pump power expressed as a multiple of the threshold power (dashed black line), both as a function of time. (b) Phase of the cavity emission, as observed with heterodyne detection. Six superradiant pulses can be seen, each with freely chosen 0 or $\pi$ phase (up to an irrelevant global phase). Shading is applied when the cavity amplitude is less than 25% of the peak value, indicated by the gray horizontal line in (a) and distinguishes between the phase locked (superradiant) and unlocked (normal) phases. (c) Histogram of phase differences between consecutive pulses, for a total of 310 experiments. The imbalance between zero versus $\pi$ is 0.3%.