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.

We directly measure the quantized collective motion of a gas of thousands of ultracold atoms, coupled to light in a high-finesse optical cavity. We detect strong asymmetries, as high as 3:1, in the intensity of light scattered into low- and high-energy motional sidebands. Owing to high cavity-atom cooperativity, the optical output of the cavity contains a spectroscopic record of the energy exchanged between light and motion, directly quantifying the heat deposited by a quantum position measurement's backaction. Such backaction selectively causes the phonon occupation of the observed collective modes to increase with the measurement rate. These results, in addition to providing a method for calibrating the motion of low-occupation mechanical systems, offer new possibilities for investigating collective modes of degenerate gases and for diagnosing optomechanical measurement backaction.

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.

A complex quantum system can be constructed by coupling simple elements. For example, trapped-ion 1,2 or superconducting 3 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 4-6 , or bidirectional coupling that is particularly strong when both objects are placed within a common electromagnetic resonator 7 . 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 8 . Here, we realize phase-coherent interactions between two addressable, spatially separated, near-groundstate 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 [8][9][10][11][12][13][14] .Cavity optomechanical systems comprised of a single mechanical oscillator interacting with a single electromagnetic cavity mode 15 serve useful quantum-mechanical functions, such as generating squeezed light [16][17][18] , detecting forces with quantum-limited sensitivity 19 or through back-action-evading measurement 20 , and both entangling and amplifying mechanical and optical modes 21 . Systems containing several mechanical elements offer additional capabilities. In the quantum regime, these systems may enable two-mode back-action-evading measurements 9 , creation of nonclassical states 10 , fundamental tests of quantum mechanics 8,11,12 , correlations at the quantum level with applications in highsensitivity measurements 13 , and quantum information science 14 . Realizing these proposed functions requires multiple-element cavity optomechanical systems in which quantum-mechanical optical force fluctuations dominate over thermal and technical ones.An important new feature in these multi-mechanical systems is photon-mediated forces between mechanical elements. Consider a driven cavity containing two mechanical elements with linear optomechanical coupling (Fig. 1a). Each element experiences radiation pressure proportional to the number of intracavity photons. The displacement of one element changes the cavity resonance frequency, causing a change in the intracavity photon number, and thereby modifying the force on the second element. In this manner, an effective optical spring is established between the mechanical elements (Fig. 1b). A quantized picture clarifies the role of cavity photons as the bidirectional force-mediating particle for this interaction: pump light is Stokes scattered off one element, generating a cavity photon that is absorbed through anti-Stokes scattering by the second element, and vice versa 7 (Fig. 1c). This cavity-mediated force has been shown to cause hybridization [22][23...

A continuous quantum field, such as a propagating beam of light, may be characterized by a squeezing spectrum that is inhomogeneous in frequency. We point out that homodyne detectors, which are commonly employed to detect quantum squeezing, are blind to squeezing spectra in which the correlation between amplitude and phase fluctuations is complex. We find theoretically that such complex squeezing is a component of ponderomotive squeezing of light through cavity optomechanics. We propose a detection scheme called synodyne detection, which reveals complex squeezing and allows the accounting of measurement backaction. Even with the optomechanical system subject to continuous measurement, such detection allows the measurement of one component of an external force with sensitivity only limited by the mechanical oscillator's thermal occupation.