Cavity-Assisted Measurement and Coherent Control of Collective Atomic Spin Oscillators

We demonstrate continuous measurement and coherent control of the collective spin of an atomic ensemble undergoing Larmor precession in a high-finesse optical cavity. The coupling of the precessing spin to the cavity field yields phenomena similar to those observed in cavity optomechanics, including cavity amplification, damping, and optical spring shifts. These effects arise from autonomous optical feedback onto the atomic spin dynamics, conditioned by the cavity spectrum. We use this feedback to stabilize the spin in either its high- or low-energy state, where, in equilibrium with measurement backaction heating, it achieves a steady-state temperature, indicated by an asymmetry between the Stokes and the anti-Stokes scattering rates. For sufficiently large Larmor frequency, such feedback stabilizes the spin ensemble in a nearly pure quantum state, in spite of continuous measurement by the cavity field.

Figure 1

(a) Schematic of the experimental system. Atoms trapped at an antinode of the cavity field experience an effective magnetic field ${\mathbf{B}}_{\mathrm{opt}}$ due to interaction with circularly polarized light. A large external magnetic field $\mathbf{B}$ along $\mathbf{x}$ defines the Larmor frequency ${\omega }_L$ and high- and low-energy stable poles of the dynamics. (b) The oscillating transverse spin component ${\widehat{F}}_z$ couples to the cavity field, causing amplitude modulation for a cavity driven either above (blue curve) or below (red curve) resonance. The optical response is delayed due to the finite cavity linewidth $\kappa$. (c) The average torque $\tau$ acting on the spin due to this optical feedback, in a rotating frame. The in-phase modulation generates torque tangential to the spin trajectory, shifting the Larmor precession frequency. The out-of-phase component generates torque perpendicular to this trajectory, causing spin nutation toward or away from the poles.

Figure 2

Coherent damping and amplification of Larmor precession of a spin ensemble with ${\omega }_L/2\pi =1.0\mathrm{MHz}$, observed in the phase modulation of transmitted light, averaged over 30–40 repetitions (blue). Cavity probe light (${\mathrm{\Delta }}_{\mathrm{pc}}/2\pi =1.0\mathrm{MHz}$, $\overline{n}=4$ average intracavity photons) drives the spin toward the high-energy pole. (a) Larmor precession of a spin ensemble, displaced from the high-energy pole by a $\pi /10$ rf pulse, coherently damps back to the pole at a rate ${\mathrm{\Gamma }}_{\mathrm{opt}}/2\pi =4.9\pm 0.2\mathrm{kHz}$. (b) A spin prepared near the low-energy pole, by application of a near $\pi$ pulse, is coherently amplified away at a rate ${\mathrm{\Gamma }}_{\mathrm{opt}}/2\pi =-4.6\pm 0.4\mathrm{kHz}$. Exponential rates are extracted by simultaneous fits (red) of both amplitude and phase quadratures. Insets show the harmonic nature of the Larmor precession signal and quality of fit in the highlighted regions. The finite cavity linewidth causes the observed signal to saturate at around 2000.

Figure 3

(a) Optical damping rates and (b) frequency shifts of Larmor precession as a function of probe detuning ${\mathrm{\Delta }}_{\mathrm{pc}}$, with fixed intracavity intensity $\overline{n}=4$ and ${\omega }_L/2\pi =1.0\mathrm{MHz}$. Diamonds (blue) label results for an ensemble initially prepared near the high-energy pole and circles (red) for an ensemble initially near the low-energy pole. Measurements repeated with either ${\sigma }^{+}$ (solid symbols) or ${\sigma }^{-}$ (open symbols) circularly polarized light demonstrate independence of optical helicity. (c) Peak damping rate as a function of Larmor frequency. Dotted vertical lines mark the value of the cavity half-linewidth $\kappa$. All theory lines are plotted with no free parameters. Error bars reflect statistical uncertainties from the fits. Additional systematic errors in the probe frequency stability and initial spin state preparation predominately affect data at small probe detuning.

Figure 4

Sideband-asymmetry thermometry of a spin oscillator. (a) Stokes and anti-Stokes sidebands observed in the averaged heterodyne power spectral density (PSD), normalized by the shot-noise (SN) PSD level, showing the sideband asymmetry of a low-energy spin, with ${\omega }_L/2\pi =900\mathrm{kHz}$, in equilibrium with an optical damping tone detuned by ${\mathrm{\Delta }}_{\mathrm{pc}}/2\pi =-1.5\mathrm{MHz}$ with average photon number ${\overline{n}}_{\mathrm{damp}}=5.0$, and measurement backaction from an on-resonance probe with average photon number ${\overline{n}}_{\text{probe}}=1.0$. (b) Inverted sideband asymmetry for a negative-temperature spin ensemble near its high-energy state, with ${\mathrm{\Delta }}_{\mathrm{pc}}/2\pi =1.5\mathrm{MHz}$, ${\overline{n}}_{\mathrm{damp}}=4.3$, and ${\overline{n}}_{\text{probe}}=2.0$. (c) Equilibrium temperature (left scale) and thermal occupation (right scale) of a high-energy spin oscillator, measured by sideband asymmetry, as a function of the Larmor frequency ${\omega }_L$. Lines indicate expected temperature at equilibrium between coherent damping and incoherent backaction. The solid line indicates the ideal limit in the absence of the on-resonance probe, and the dashed line includes the additional backaction for the average probe cooperativity.