Negative-Mass Instability of the Spin and Motion of an Atomic Gas Driven by Optical Cavity Backaction

We realize a spin-orbit interaction between the collective spin precession and center-of-mass motion of a trapped ultracold atomic gas, mediated by spin- and position-dependent dispersive coupling to a driven optical cavity. The collective spin, precessing near its highest-energy state in an applied magnetic field, can be approximated as a negative-mass harmonic oscillator. When the Larmor precession and mechanical motion are nearly resonant, cavity mediated coupling leads to a negative-mass instability, driving exponential growth of a correlated mode of the hybrid system. We observe this growth imprinted on modulations of the cavity field and estimate the full covariance of the resulting two-mode state by observing its transient decay during subsequent free evolution.

Figure 1

(a),(b) Energy levels of two nearly degenerate harmonic oscillators. (a) For positive-mass oscillators, coupling mediates exchange of excitations, conserving the total excitation number. (b) For positive- and negative-mass oscillators, the interaction resonantly drives pair creation, resulting in exponential growth of a correlated mode. (c) The center-of-mass motion of a harmonically confined, ultracold atomic ensemble (green), with trap frequency ${\omega }_m$, represents the positive-mass oscillator. Larmor precession at frequency ${\omega }_s$ of the collective atomic spin near its highest-energy state, in an applied magnetic field $\mathbf{B}\propto \mathbf{x}$, approximates a negative-mass oscillator. Position- and spin-dependent dispersive coupling to a circularly polarized mode of the optical cavity (red) mediates coherent interaction between the oscillators and facilitates continuous measurement of their dynamics.

Figure 2

Observation of negative-mass instability. (a) Spectrogram of total optical modulation observed during the experimental sequence, averaged over 200 iterations. The spectrum shows components at the Larmor frequency, fixed at ${\omega }_s=120\mathrm{kHz}$ (dashed line), and the mechanical frequency, initially at ${\omega }_m=95\mathrm{kHz}$, then varied during coupling to achieve the desired detuning (solid line). The collective spin shows negligible decay and the motion damps at rate ${\mathrm{\Gamma }}_m/2\pi =2\mathrm{kHz}$. (b) Optodynamical coupling strength ${\mathrm{\Omega }}_{\text{opt}}$, calculated from the measured ${\mathrm{\Delta }}_{\text{pc}}$ and $\overline{n}$. Experiments are performed with a coupling pulse with average $\overline{n}=15$ and ${\mathrm{\Delta }}_{\text{pc}}=1.4\mathrm{MHz}$ (blue) and without coupling (red). (c) The mean squared joint displacement of both oscillators, captured in the cycle-averaged optical modulation power between 85 and 150 kHz. This signal reflects exponential amplification of both oscillators while coupled, followed by a stationary beat during the subsequent free evolution, revealing the transient decay of correlations created between the two modes. Transients from changes in the optical probe and trap intensity perturb measurements near $t=0$ and $t=t_c$ (light points), which are excluded from analysis.

Figure 3

(a) Onset of instability, observed in the mean squared displacement, for increasing optical coupling strength, with $\overline{n}=10$ and oscillator detuning $\delta /2\pi =14\mathrm{kHz}$. Each trace (offset for clarity) is the average of around 30 repetitions. Growth saturates due to the finite cavity linewidth (scale bar) and other nonlinearities. (b) Resonance of instability for varied $\delta$, under strongest optical coupling in (a). The instability gain $G_{+}$ is extracted by least-square fits (lines) to data at early times. (c) $G_{+}$ (red diamonds) versus $\delta$, compared with predicted steady-state gain (solid line). The peak instability occurs at finite detuning, due to optodynamical shifts of each oscillator’s frequency. The larger frequency shift observed might be due to asymmetric transients, not reflected in the theoretical steady-state gain. (d) For an inverted probe detuning (${\mathrm{\Delta }}_{\text{pc}}=+2.0\mathrm{MHz}$), the optodynamical coupling acts opposite the static coupling, resulting in reduced peak gain (blue diamonds). Error bars in (c)–(d) represent combined $1\text{−}\sigma$ statistical uncertainty from the fit and systematic error estimated from $\pm 10\%$ variations of the fit interval.

Figure 4

Results of matched-filter analysis. (a) Growth of the mechanical (red downward triangle), spin (blue upward triangle), and correlated (yellow squares) occupation as a function of $t_c$, with $\delta /2\pi =14\mathrm{kHz}$ and the same optical coupling as Fig. 3. The observed correlated occupation agrees well with the predicted evolution of the measured initial state at the optimal detuning (yellow line). Growth saturates near a mechanical occupation of 100, possibly due to mechanical nonlinearity of the dipole trap. (b) Correlated occupation and phase versus $\delta$, after fixed $t_c=60\mu \mathrm{s}$. Error bars in (a)–(b) indicate combined $1\text{−}\sigma$ statistical uncertainty after 200 repetitions and estimated systematic error from the uncertainties of all filter parameters. The anomalous frequency shift is similar to that seen in Figs. 3 and 3. (c) Comparison between the beat observed in $⟨{\widehat{D}}^2{⟩}_{\mathrm{cyc}}$ (blue points) and a simulated signal constructed from time evolution of the estimated two-mode state (black line) for one typical experimental setting. Shaded regions show $1\text{−}\sigma$ bounds for evolution of a maximally-correlated state with the same estimated individual oscillator occupations. Similar measurements performed without coupling (red points) show evolution of the initial state.