Laser cooling of a nanomechanical oscillator into its quantum ground state

Abstract

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

Cooling-by-measurement and mechanical state tomography via pulsed optomechanics

“…A versatile approach to manipulate mechanical states of motion is provided by the interaction with electromagnetic radiation, typically confined to microwave or optical cavities. Such cavity-optomechanics experiments [4][5][6][7][8] have thus far largely concentrated on high sensitivity continuous monitoring of the mechanical position [9][10][11][12][13][14]. Because of the back-action imparted by the probe onto the measured object, the precision of such a measurement is fundamentally constrained by the standard quantum limit (SQL) [15,16], and therefore only allows for classical phase-space reconstruction [9,17,18].…”

Quantum Limits on Measurement and Control of a Mechanical Oscillator

“…Such dynamic backaction on the mechanical oscillator leading to optical spring [198,199], linewidth narrowing (amplification) [200][201][202] or damping (cooling) [203][204][205][206][207] have been observed. In the resolvedsideband regime, characterised by κ Ω m , it becomes possible to cool the mechanical oscillator to a level where its energy is comparable to the energy in its ground state [208][209][210][211].…”

“…Coupling of solid state mechanical oscillators to an optical cavity mode serving as a cold bath -effectively realizing an autonomous feedback loop (i.e. dynamic back-action) -has enabled thermal noise reduction to the level of the zero-point motion [209][210][211]. In this case, back-action imposes a fundamental limit [310] which may be mitigated by operating in an appropriate parameter regime (the resolvedsideband regime [311,312]).…”

Multichannel cavity optomechanics for all-optical amplification of radio frequency signals

“…Using high finesse optical cavities, optomechanical interaction can be resonantly enhanced. Research in cavity optomechanics exploits the dynamical interplay between the intra-cavity optical field and the mechanical motion of the device, leading to demonstrations of unprecedented phenomena including backaction cooling [10][11][12][13] , normal mode splitting 14 and optomechanically induced transparency 15,16 . Other than fundamental studies, a plethora of promising applications of cavity optomechanics, including tunable photonic filters and wavelength router 5,6 , wavelength conversion and switching 8,9 , radio-frequency optomechanical oscillators 17,18 and non-volatile optical memory 19 , have emerged.…”