Resolved-sideband cooling and measurement of a micromechanical oscillator close to the quantum limit

The observation of quantum phenomena in macroscopic mechanical oscillators has been a subject of interest since the inception of quantum mechanics. Prerequisite to this regime are both preparation of the mechanical oscillator at low phonon occupancy and a measurement sensitivity at the scale of the spread of the oscillator's ground state wavefunction. It has been widely perceived that the most promising approach to address these two challenges are electro nanomechanical systems. Here we approach for the first time the quantum regime with a mechanical oscillator of mesoscopic dimensions--discernible to the bare eye--and 1000-times more massive than the heaviest nano-mechanical oscillators used to date. Imperative to these advances are two key principles of cavity optomechanics: Optical interferometric measurement of mechanical displacement at the attometer level, and the ability to use measurement induced dynamic back-action to achieve resolved sideband laser cooling of the mechanical degree of freedom. Using only modest cryogenic pre-cooling to 1.65 K, preparation of a mechanical oscillator close to its quantum ground state (63+-20 phonons) is demonstrated. Simultaneously, a readout sensitivity that is within a factor of 5.5+-1.5 of the standard quantum limit is achieved. The reported experiments mark a paradigm shift in the approach to the quantum limit of mechanical oscillators using optical techniques and represent a first step into a new era of experimental investigation which probes the quantum nature of the most tangible harmonic oscillator: a mechanical vibration.Comment: 14 pages, 4 figure

Determination of the vacuum optomechanical coupling rate using frequency noise calibration

The strength of optomechanical interactions in a cavity optomechanical system can be quantified by a vacuum coupling rate \vcr analogous to cavity quantum electrodynamics. This single figure of merit removes the ambiguity in the frequently quoted coupling parameter defining the frequency shift for a given mechanical displacement, and the effective mass of the mechanical mode. Here we demonstrate and verify a straightforward experimental technique to derive the vacuum optomechanical coupling rate. It only requires applying a known frequency modulation of the employed electromagnetic probe field and knowledge of the mechanical oscillator's occupation. The method is experimentally verified for a micromechanical mode in a toroidal whispering-gallery-resonator and a nanomechanical oscillator coupled to a toroidal cavity via its near field.Comment: 11 pages, 2 figure

From Cavity Electromechanics to Cavity Optomechanics

We present an overview of experimental work to embed high-Q mesoscopic mechanical oscillators in microwave and optical cavities. Based upon recent progress, the prospect for a broad field of "cavity quantum mechanics" is very real. These systems introduce mesoscopic mechanical oscillators as a new quantum resource and also inherently couple their motion to photons throughout the electromagnetic spectrum.Comment: 8 pages, 6 figures, ICAP proceedings submissio

High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators

The inherent coupling of optical and mechanical modes in high finesse optical microresonators provide a natural, highly sensitive transduction mechanism for micromechanical vibrations. Using homodyne and polarization spectroscopy techniques, we achieve shot-noise limited displacement sensitivities of 10^(-19) m Hz^(-1/2). In an unprecedented manner, this enables the detection and study of a variety of mechanical modes, which are identified as radial breathing, flexural and torsional modes using 3-dimensional finite element modelling. Furthermore, a broadband equivalent displacement noise is measured and found to agree well with models for thermorefractive noise in silica dielectric cavities. Implications for ground-state cooling, displacement sensing and Kerr squeezing are discussed.Comment: 25 pages, 8 figure

On optical forces in spherical whispering gallery mode resonators

In this paper we discuss the force exerted by the field of an optical cavity on a polarizable dipole. We show that the modification of the cavity modes due to interaction with the dipole significantly alters the properties of the force. In particular, all components of the force are found to be non-conservative, and cannot, therefore, be derived from a potential energy. We also suggest a simple generalization of the standard formulas for the optical force on the dipole, which reproduces the results of calculations based on the Maxwell stress tensor.Comment: To pe published in Optics Express Focus Issue: "Collective phenomena in photonic, plasmonic and hybrid structures

Measuring nanomechanical motion with an imprecision far below the standard quantum limit

We demonstrate a transducer of nanomechanical motion based on cavity enhanced optical near-fields capable of achieving a shot-noise limited imprecision more than 10 dB below the standard quantum limit (SQL). Residual background due to fundamental thermodynamical frequency fluctuations allows a total imprecision 3 dB below the SQL at room temperature (corresponding to 600 am/Hz^(1/2) in absolute units) and is known to reduce to negligible values for moderate cryogenic temperatures. The transducer operates deeply in the quantum backaction dominated regime, prerequisite for exploring quantum backaction, measurement-induced squeezing and accessing sub-SQL sensitivity using backaction evading techniques

Local Optical Probe of Motion and Stress in a multilayer graphene NEMS

Nanoelectromechanical systems (NEMSs) are emerging nanoscale elements at the crossroads between mechanics, optics and electronics, with significant potential for actuation and sensing applications. The reduction of dimensions compared to their micronic counterparts brings new effects including sensitivity to very low mass, resonant frequencies in the radiofrequency range, mechanical non-linearities and observation of quantum mechanical effects. An important issue of NEMS is the understanding of fundamental physical properties conditioning dissipation mechanisms, known to limit mechanical quality factors and to induce aging due to material degradation. There is a need for detection methods tailored for these systems which allow probing motion and stress at the nanometer scale. Here, we show a non-invasive local optical probe for the quantitative measurement of motion and stress within a multilayer graphene NEMS provided by a combination of Fizeau interferences, Raman spectroscopy and electrostatically actuated mirror. Interferometry provides a calibrated measurement of the motion, resulting from an actuation ranging from a quasi-static load up to the mechanical resonance while Raman spectroscopy allows a purely spectral detection of mechanical resonance at the nanoscale. Such spectroscopic detection reveals the coupling between a strained nano-resonator and the energy of an inelastically scattered photon, and thus offers a new approach for optomechanics

Suppression of extraneous thermal noise in cavity optomechanics

Extraneous thermal motion can limit displacement sensitivity and radiation pressure effects, such as optical cooling, in a cavity-optomechanical system. Here we present an active noise suppression scheme and its experimental implementation. The main challenge is to selectively sense and suppress extraneous thermal noise without affecting motion of the oscillator. Our solution is to monitor two modes of the optical cavity, each with different sensitivity to the oscillator's motion but similar sensitivity to the extraneous thermal motion. This information is used to imprint "anti-noise" onto the frequency of the incident laser field. In our system, based on a nano-mechanical membrane coupled to a Fabry-P\'{e}rot cavity, simulation and experiment demonstrate that extraneous thermal noise can be selectively suppressed and that the associated limit on optical cooling can be reduced.Comment: 27 pages, 14 figure

Horizontal Versus Vertical Fiscal Equalization

We analyze a model in which the provision of regional public goods by regional governments leads to spillover effects and in whi ch the central government can establish a vertical equalization scheme while the regional governments can set up a horizontal equalization scheme. The two levels of government decide in different chronological order. It turns out that, regardless of the ti ming, the central government always prevails. Horizontal equalization does not take place – nor is it necessary in order to achieve constrain ed Pareto efficiency. Moreover, if in the model economy the goal of achieving equality in living conditions across the regions is pursued, the only suitable candidate for reaching this goal is vertical equalization

A microchip optomechanical accelerometer

The monitoring of accelerations is essential for a variety of applications ranging from inertial navigation to consumer electronics. The basic operation principle of an accelerometer is to measure the displacement of a flexibly mounted test mass; sensitive displacement measurement can be realized using capacitive, piezo-electric, tunnel-current, or optical methods. While optical readout provides superior displacement resolution and resilience to electromagnetic interference, current optical accelerometers either do not allow for chip-scale integration or require bulky test masses. Here we demonstrate an optomechanical accelerometer that employs ultra-sensitive all-optical displacement read-out using a planar photonic crystal cavity monolithically integrated with a nano-tethered test mass of high mechanical Q-factor. This device architecture allows for full on-chip integration and achieves a broadband acceleration resolution of 10 \mu g/rt-Hz, a bandwidth greater than 20 kHz, and a dynamic range of 50 dB with sub-milliwatt optical power requirements. Moreover, the nano-gram test masses used here allow for optomechanical back-action in the form of cooling or the optical spring effect, setting the stage for a new class of motional sensors.Comment: 16 pages, 9 figure