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

THEORY OF OPTOMECHANICAL EIT, EIA AND PARAMETRIC AMPLIFICATIONHere we provide a theoretical treatment of some of the main aspects of EIT [1][2][3][4], EIA [5] and parametric amplification [6][7][8] in optomechanical systems. Modeling the optomechanical system with the Hamiltonianit is possible to linearize the operation of the system, under the influence of a control laser at ω c , about a particular steady-state given by intracavity photon amplitude α 0 and a static phonon shift β 0 . The interaction of the mechanics and pump photons at ω c with secondary "probe" photons at ω s = ω c ± ∆ with two-photon detuning ∆ can then be modeled by making the substitutionŝAssuming that the pump is much larger than the probe, |α 0 | |α ± |, the pump amplitude is left unaffected and the equations for each sideband amplitude α ± are found to beWe have defined ∆ OC = ω o − ω c as the pump detuning from the optical cavity (including the static optomechanical shift, ω o ), and β + = β * − . In these situations it is typical to define G = gα 0 , as the effective optomechanical coupling rate between a sideband and the mechanical subsystem, mediated by the pump. Red-detuned pump: Electromagnetically Induced TransparencyWith the pump detuned from the cavity by a two-photon detuning ∆, the spectral selectivity of the optical cavity causes the sideband populations to be skewed in a drastic fashion. It is then an acceptable approximation to neglect one of these sidebands, depending on whether the pump is on the red or blue side of the cavity. When the pump resides on the red side (∆ OC > 0), the α + is reduced and can be neglected. This is the rotating wave approximation (RWA) and is valid so long as ∆ κ. Then Eqs. (S3-S4) may be solved for the reflection and transmission coefficients r(ω s ) and t(ω s ) of the side-coupled cavity system. We find that These equations are plotted in Figs. S1 and S2.

In this Letter we use resolved sideband laser cooling to cool a mesoscopic mechanical resonator to near its quantum ground state (phonon occupancy 2:6 AE 0:2), and observe the motional sidebands generated on a second probe laser. Asymmetry in the sideband amplitudes provides a direct measure of the displacement noise power associated with quantum zero-point fluctuations of the nanomechanical resonator, and allows for an intrinsic calibration of the phonon occupation number. DOI: 10.1103/PhysRevLett.108.033602 PACS numbers: 42.50.Wk, 42.65.Àk, 62.25.Àg Experiments with trapped ions and neutral atoms [1-3], dating back several decades, utilized techniques such as resolved sideband laser cooling and motional sideband absorption and fluorescence spectroscopy to cool and measure a single trapped particle in its vibrational quantum ground state. These experiments generated significant interest in the coherent control of motion and the quantum optics of trapped atoms and ions [4], and were important stepping stones towards the development of ion-trap based quantum computing [5,6]. Larger scale mechanical objects, such as fabricated nanomechanical resonators, have only recently been cooled close to their quantum mechanical ground state of motion [7][8][9][10][11][12][13][14]. In a pioneering experiment by O'Connell, et al.[11], a piezoelectric nanomechanical resonator has been cryogenically cooled (T b $ 25 mK) to its vibrational ground state and strongly coupled to a superconducting circuit qubit allowing for quantum state preparation and readout of the mechanics. An alternate line of research has been pursued in circuit and cavity optomechanics [15], where the position of a mechanical oscillator is coupled to the frequency of a high-Q electromagnetic resonance allowing for backaction cooling [16,17] and continuous position readout of the oscillator. Such optomechanical resonators have long been pursued as quantum-limited sensors of weak classical forces [9,15,[18][19][20], with more recent studies exploring optomechanical systems as quantum optical memories and amplifiers [21][22][23][24], quantum nonlinear dynamical elements [25], and quantum interfaces in hybrid quantum systems [26][27][28][29].Despite the major advances in circuit and cavity optomechanical systems made in the last few years, all experiments to date involving the cooling of mesoscopic mechanical oscillators have relied on careful measurement and calibration of the motion-induced scattering of light to obtain the average phonon occupancy of the oscillator, hni. Approach towards the quantum ground state in such experiments is manifest only as a weaker measured signal, with no evident demarcation between the classical and quantum regimes of the oscillator. A crucial aspect of zero-point fluctuations (zpfs) of the quantum ground state is that they cannot supply energy, but can only contribute to processes where energy is absorbed by the mechanics. This is different from classical noise, and techniques that attempt to measure zero-point motion withou...

The nitrogen vacancy ͑NV͒ center in diamond is promising as an electron spin qubit due to its long-lived coherence and optical addressability. The ground state is a spin triplet with two levels ͑m s = ±1͒ degenerate at zero magnetic field. Polarization-selective microwave excitation is an attractive method to address the spin transitions independently since this allows operation down to zero magnetic field. Using a resonator designed to produce circularly polarized microwaves, we have investigated the polarization selection rules of the NV center. We first apply this technique to NV ensembles in ͓100͔-and ͓111͔-oriented samples. Next, we demonstrate an imaging technique, based on optical polarization dependence, which allows rapid identification of the orientations of many single NV centers. Finally, we test the microwave polarization selection rules of individual NV centers of known orientation.