DERIVATION OF TRANSDUCED SIGNALWe begin by modeling the optomechanical system with the Hamiltonian H =hΔâ †â +hω mb †b +hg(b † +b)â †â + ih κ e 2 α in,where Δ = ω o − ω l , with laser frequency ω l , optical mode frequency ω o and mechanical mode frequency ω m . Herê a (â † ) andb (b † ) are respectively the annihilation (creation) operators of photon and phonon resonator quanta, g is the optomechanical coupling rate corresponding physically to the shift in the optical mode frequency due to the zero-point fluctuations (x zpf = h/2mω m , m motional mass) of the phonon mode. Classical Derivation of Observed SpectraBy making the substitutionsâwe can treat the system classically by representing the photon amplitudes as a Fourier decomposition of sidebands. Notice that the infinite summation over each sideband order q, can be relaxed to a few orders in the sideband resolved regime (κ ω m ). The phonon amplitude, β 0 , is the classical mechanical excitation amplitude. For an oscillator undergoing thermal Brownian motion, β 0 , is a stochastic process. We assert the stochastic nature of the variable, at the end of the derivation where the power spectral density is calculated. The equation of motion for the slowly varying component is thenwhere we introduce the cavity (optical) energy loss rate, κ, and the cavity coupling rate, κ e . This can be written as a system of equations M · α = a in whereBy truncating and inverting the coupling matrix M one can determine each one of the sidebands amplitude as α q = (M −1 ) qp a in,p and therefore determine the steady state power leaving the cavity to beSUPPLEMENTARY INFORMATION

Achieving coherent quantum control over massive mechanical resonators is a current research goal. Nano- and micromechanical devices can be coupled to a variety of systems, for example to single electrons by electrostatic or magnetic coupling, and to photons by radiation pressure or optical dipole forces. So far, all such experiments have operated in a regime of weak coupling, in which reversible energy exchange between the mechanical device and its coupled partner is suppressed by fast decoherence of the individual systems to their local environments. Controlled quantum experiments are in principle not possible in such a regime, but instead require strong coupling. So far, this has been demonstrated only between microscopic quantum systems, such as atoms and photons (in the context of cavity quantum electrodynamics) or solid state qubits and photons. Strong coupling is an essential requirement for the preparation of mechanical quantum states, such as squeezed or entangled states, and also for using mechanical resonators in the context of quantum information processing, for example, as quantum transducers. Here we report the observation of optomechanical normal mode splitting, which provides unambiguous evidence for strong coupling of cavity photons to a mechanical resonator. This paves the way towards full quantum optical control of nano- and micromechanical devices.

Most working scientists hold fast to the concept of 'realism'--a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell's theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of 'spooky' actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.

We present the measurement of squeezed light generation using an engineered optomechanical system fabricated from a silicon microchip and composed of a micromechanical resonator coupled to a nanophotonic cavity. Laser light is used to measure the fluctuations in the position of the mechanical resonator at a measurement rate comparable to the free dynamics of the mechanical resonator, and greater than its thermal decoherence rate. By approaching the strong continuous measurement regime we observe, through homodyne detection, non-trivial modifications of the reflected light's vacuum fluctuation spectrum. In spite of the mechanical resonator's highly excited thermal state (10, 000 phonons), we observe squeezing at the level of 4.5 ± 0.5% below that of shot-noise over a few MHz bandwidth around the mechanical resonance frequency of 28 MHz. This squeezing is interpreted as an unambiguous quantum signature of radiation pressure shot-noise.Monitoring a mechanical object's motion, even with the gentle touch of light, fundamentally alters its dynamics. The experimental manifestation of this basic principle of quantum mechanics, its link to the quantum nature of light, and the extension of quantum measurement to the macroscopic realm have all received extensive attention over the last half century [1,2]. The use of squeezed light, with quantum fluctuations below that of the vacuum field, was proposed nearly three decades ago [3] as a means for beating the standard quantum limits in precision displacement and force measurements. Conversely, it has also been proposed that a strong continuous measurement of a mirror's position with light, may itself give rise to squeezed light [4,5]. In this Letter, we present such a continuous position measurement using an engineered optomechanical system fabricated from a silicon microchip and composed of a micromechanical resonator coupled to a nanophotonic cavity. Laser light is used to measure the fluctuations in the position of the mechanical resonator at a measurement rate comparable to the free dynamics of the mechanical resonator, and greater than its thermal decoherence rate. By approaching the strong continuous measurement regime we observe, through homodyne detection, non-trivial modifications of the reflected light's vacuum fluctuation spectrum. In spite of the mechanical resonator's highly excited thermal state (10 4 phonons), we observe squeezing at the level of 4.5 ± 0.5% below that of shot-noise over a few MHz bandwidth around the mechanical resonance frequency of ω m /2π ≈ 28 MHz. This * These authors contributed equally to this work.† Electronic address: opainter@caltech.edu; URL: http://copilot. caltech.edu squeezing is interpreted as an unambiguous quantum signature of radiation pressure shot-noise [6]. The generation of states of light with fluctuations below that of vacuum has been of great theoretical interest since the 1970s [3,[7][8][9]. Early experimental work demonstrated squeezing of a few percent in a large variety of different nonlinear systems, such as neutral...

Entanglement, an essential feature of quantum theory that allows for inseparable quantum correlations to be shared between distant parties, is a crucial resource for quantum networks . Of particular importance is the ability to distribute entanglement between remote objects that can also serve as quantum memories. This has been previously realized using systems such as warm and cold atomic vapours, individual atoms and ions, and defects in solid-state systems. Practical communication applications require a combination of several advantageous features, such as a particular operating wavelength, high bandwidth and long memory lifetimes. Here we introduce a purely micromachined solid-state platform in the form of chip-based optomechanical resonators made of nanostructured silicon beams. We create and demonstrate entanglement between two micromechanical oscillators across two chips that are separated by 20 centimetres . The entangled quantum state is distributed by an optical field at a designed wavelength near 1,550 nanometres. Therefore, our system can be directly incorporated in a realistic fibre-optic quantum network operating in the conventional optical telecommunication band. Our results are an important step towards the development of large-area quantum networks based on silicon photonics.

Interfacing a single photon with another quantum system is a key capability in modern quantum information science. It allows quantum states of matter, such as spin states of atoms [1,2], atomic ensembles [3,4] or solids [5], to be prepared and manipulated by photon counting and, in particular, to be distributed over long distances. Such light-matter interfaces have become crucial to fundamental tests of quantum physics [6] and realizations of quantum networks [7]. Here we report non-classical correlations between single photons and phonons -the quanta of mechanical motion -from a nanomechanical resonator. We implement a full quantu protocol involving initialization of the resonator in its quantum ground state of motion and subsequent generation and read-out of correlated photon-phonon pairs. The observed violation of a Cauchy-Schwarz inequality is clear evidence for the non-classical nature of the mechanical state generated. Our results demonstrate the availability of on-chip solid-state mechanical resonators as light-matter quantum interfaces. The performance we achieved will enable studies of macroscopic quantum phenomena [8] as well as applications in quantum communication [9], as quantum memories [10] and as quantum transducers [11,12].Over the past few years, nanomechanical devices have been discussed as possible building blocks for quantum information architectures [9,13]. Their unique feature is that they combine an engineerable solid-state platform on the nanoscale with the possibility to coherently interact with a variety of physical quantum systems including electronic or nuclear spins, single charges, and photons [14,15]. This feature enables mechanics-based hybrid quantum systems that interconnect different, independent physical qubits through mechanical modes.A successful implementation of such quantum transducers requires the ability to create and control quantum states of mechanical motion. The first step -the initialization of micro-and nanomechanical systems in their quantum ground state of motion -has been realized in various mechanical systems either through direct cryogenic cooling [16,17] or laser cooling using microwave [18] and optical cavity fields [19]. Further progress in quantum state control has mainly been limited to the domain of electromechanical devices, in which mechanical motion couples to superconducting circuits in the form of qubits and microwave cavities [15]. Recent achievements include single-phonon control of a micromechanical resonator by a superconducting flux qubit [16], the generation of quantum entanglement between quadratures of a microwave cavity field and micromechanical motion [20], * This work was published in Nature 530, 313-316 (2016 Interfacing mechanics with optical photons in the quantum regime is highly desirable because it adds important features such as the ability to transfer mechanical excitations over long distances [9,24]. In addition, the available toolbox of single-photon generation and detection allows for remote quantum state control [7]. However...

All quantum optomechanics experiments to date operate at cryogenic temperatures, imposing severe technical challenges and fundamental constraints. Here we present a novel design of onchip mechanical resonators which exhibit fundamental modes with frequencies f and mechanical quality factors Qm sufficient to enter the optomechanical quantum regime at room temperature. We overcome previous limitations by designing ultrathin, high-stress silicon nitride (Si3N4) membranes, with tensile stress in the resonators' clamps close to the ultimate yield strength of the material. By patterning a photonic crystal on the SiN membranes, we observe reflectivities greater than 99%. These on-chip resonators have remarkably low mechanical dissipation, with Qm∼10 8 , while at the same time exhibiting large reflectivities. This makes them a unique platform for experiments towards the observation of massive quantum behavior at room temperature.

We produce two identical keys using, for the first time, entangled trinary quantum systems (qutrits) for quantum key distribution. The advantage of qutrits over the normally used binary quantum systems is an increased coding density and a higher security margin. The qutrits are encoded into the orbital angular momentum of photons, namely Laguerre-Gaussian modes with azimuthal index l +1, 0 and −1, respectively. The orbital angular momentum is controlled with phase holograms. In an Ekert-type protocol the violation of a three-dimensional Bell inequality verifies the security of the generated keys. A key is obtained with a qutrit error rate of approximately 10%.