The mean-field treatment of the Bose-Hubbard model predicts properties of lattice-trapped gases to be insensitive to the specific lattice geometry once system energies are scaled by the lattice coordination number z. We test this scaling directly by comparing coherence properties of 87Rb gases that are driven across the superfluid to Mott insulator transition within optical lattices of either the kagome (z = 4) or the triangular (z = 6) geometries. The coherent fraction measured for atoms in the kagome lattice is lower than for those in a triangular lattice with the same interaction and tunneling energies. A comparison of measurements from both lattices agrees quantitatively with the scaling prediction.

# 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 back-action heating, it achieves a steady-state temperature, indicated by an asymmetry between the Stokes and 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.

# Cavity-mediated coupling of mechanical oscillators limited by quantum backaction

A complex quantum system can be constructed by coupling simple elements. For example, trapped-ion, or superconducting quantum bits may be coupled by Coulomb interactions, mediated by the exchange of virtual photons. Alternatively, quantum objects can be made to emit and exchange real photons, providing either unidirectional coupling in cascaded geometries, or bidirectional coupling that is particularly strong when both objects are placed within a common electromagnetic resonator. However, in such an open system, the capacity of a coupling channel to convey quantum information or generate entanglement may be compromised by photon loss. Here, we realize phase-coherent interactions between two addressable, spatially separated, near-ground-state mechanical oscillators within a driven optical cavity. We observe the quantum back-action noise imparted by the optical coupling resulting in correlated mechanical fluctuations of the two oscillators. Our results illustrate challenges and opportunities of coupling quantum objects with light for applications of quantum cavity optomechanics.

# Cooling a Bose gas to 0.02 times the condensation temperature

Trapped quantum gases can be cooled to impressively low temperatures, but it is unclear whether their entropy is low enough to realize phenomena such as d-wave superconductivity and magnetic ordering. Estimated critical entropies per particle for quantum magnetic ordering are \( \sim 0.3k_B\) and \( \sim 0.03k_B\) for bosons in three- and two-dimensional lattices, respectively, with similar values for Néel ordering of lattice-trapped Fermi gases. Here we report reliable single-shot temperature measurements of a degenerate Rb gas by imaging the momentum distribution of thermalized magnons, which are spin excitations of the atomic gas. We record average temperatures fifty times lower than the Bose–Einstein condensation temperature, indicating an entropy per particle of \( \sim 0.001 k_B\) at equilibrium, nearly two orders of magnitude lower than the previous best in a dilute atomic gas, and well below the critical entropy for antiferromagnetic ordering of a Bose–Hubbard system. The magnons can reduce the temperature of the system by absorbing energy during thermalization and by enhancing evaporative cooling, allowing the production of low-entropy gases in deep traps.

# Collective excitation interferometry with a toroidal BEC

The precision of most compact inertial sensing schemes using trapped- and guided-atom interferometers has been limited by uncontrolled phase errors caused by trapping potentials and interactions. Here we propose an acoustic interferometer that uses sound waves in a toroidal Bose-Einstein condensate to measure rotation, and we demonstrate experimentally several key aspects of this type of interferometer. We use spatially patterned light beams to excite counterpropagating sound waves within the condensate and use *in situ* absorption imaging to characterize their evolution. We present an analysis technique by which we extract separately the oscillation frequencies of the standing-wave acoustic modes, the frequency splitting caused by static imperfections in the trapping potential, and the characteristic precession of the standing-wave pattern due to rotation. Supported by analytic and numerical calculations, we interpret the noise in our measurements, which is dominated by atom shot noise, in terms of rotation noise. While the noise of our acoustic interferometric sensor, at the level of \(\sim \text{rad s}^{−1}/\sqrt{\text{Hz}} \), is high owing to rapid acoustic damping and the small radius of the trap, the proof-of-concept device does operate at the high densities and small volumes of trapped Bose-Einstein condensed gases.

# Coherent Magnon Optics in a Ferromagnetic Spinor BEC

We measure the dispersion relation, gap, and magnetic moment of a magnon in the ferromagnetic \( F=1 \) spinor Bose-Einstein condensate of \( ^{87}\text{Rb} \). From the dispersion relation we measure an average effective mass \( 1.033(2)_\text{stat}(10)_\text{sys} \) times the atomic mass, as determined by interfering standing and running coherent magnon waves within the dense and trapped condensed gas. The measured mass is higher than theoretical predictions of mean-field and beyond-mean-field Beliaev theory for a bulk spinor Bose gas with \(s\)-wave contact interactions. We observe a magnon energy gap of \(h \times 2.5(1)_\text{stat}(2)_\text{sys} \text{Hz} \), which is consistent with the predicted effect of magnetic dipole-dipole interactions. These dipolar interactions may also account for the high magnon mass. The effective magnetic moment of \( −1.04(2)_\text{stat}(8)_\text{sys} \) times the atomic magnetic moment is consistent with mean-field theory.

# Standard quantum limit of force sensitivity

The Heisenberg uncertainty principle sets a lower bound on the noise in a force measurement based on continuously detecting a mechanical oscillator’s position. This bound, the standard quantum limit, can be reached when the oscillator subjected to the force is unperturbed by its environment and when measurement imprecision from photon shot noise is balanced against disturbance from measurement back-action. We applied an external force to the center-of-mass motion of an ultracold atom cloud in a high-finesse optical cavity and measured the resulting motion optically. When the driving force is resonant with the cloud’s oscillation frequency, we achieve a sensitivity that is a factor of 4 above the standard quantum limit and consistent with theoretical predictions given the atoms’ residual thermal disturbance and the photodetection quantum efficiency.

# Observation of ponderomotive squeezing

Optomechanical systems, in which light drives and is affected by the motion of a massive object, will comprise a new framework for nonlinear quantum optics, with applications ranging from the storage and transduction of quantum information to enhanced detection sensitivity in gravitational wave detectors. However, quantum optical effects in optomechanical systems have remained obscure, because their detection requires the object’s motion to be dominated by vacuum fluctuations in the optical radiation pressure; so far, direct observations have been stymied by technical and thermal noise. Here we report an implementation of cavity optomechanics using ultracold atoms in which the collective atomic motion is dominantly driven by quantum fluctuations in radiation pressure. The back-action of this motion onto the cavity light field produces ponderomotive squeezing. We detect this quantum phenomenon by measuring sub-shot-noise optical squeezing. Furthermore, the system acts as a low-power, high-gain, nonlinear parametric amplifier for optical fluctuations, demonstrating a gain of 20 dB with a pump corresponding to an average of only seven intracavity photons. These findings may pave the way for low-power quantum optical devices, surpassing quantum limits on position and force sensing, and the control and measurement of motion in quantum gases.