We report on a stable optical trap suitable for a macroscopic mirror, wherein
the dynamics of the mirror are fully dominated by radiation pressure. The
technique employs two frequency-offset laser fields to simultaneously create a
stiff optical restoring force and a viscous optical damping force. We show how
these forces may be used to optically trap a free mass without introducing
thermal noise; and we demonstrate the technique experimentally with a 1 gram
mirror. The observed optical spring has an inferred Young's modulus of 1.2 TPa,
20% stiffer than diamond. The trap is intrinsically cold and reaches an
effective temperature of 0.8 K, limited by technical noise in our apparatus.Comment: Major revision. Replacement is version that appears in Phy. Rev.
  Lett. 98, 150802 (2007

Chen, Y.

Corbitt, T.

Innerhofer, E.

Mavalvala, N.

Mueller-Ebhardt, H.

Ottaway, D.

Rehbein, H.

Sigg, D.

Whitcomb, S.

Wipf, C.

Physical Review Letters

English

arXiv

We report on a stable optical trap suitable for a macroscopic mirror, wherein the dynamics of the mirror are fully dominated by radiation pressure. The technique employs two frequency-offset laser fields to simultaneously create a stiff optical restoring force and a viscous optical damping force. We show how these forces may be used to optically trap a free mass without introducing thermal noise, and we demonstrate the technique experimentally with a 1 g mirror. The observed optical spring has an inferred Young's modulus of 1.2 TPa, 20% stiffer than diamond. The trap is intrinsically cold and reaches an effective temperature of 0.8 K, limited by technical noise in our apparatus.Thomas Corbitt, Yanbei Chen, Edith Innerhofer, Helge Müller-Ebhardt, David Ottaway, Henning Rehbein, Daniel Sigg, Stanley Whitcomb, Christopher Wipf, and Nergis Mavalval

Muller-Ebhardt, H.

Adelaide Research & Scholarship

An all-optical trap for a gram-scale mirror

We report on a stable optical trap suitable for a macroscopic mirror, wherein the dynamics of the mirror are fully dominated by radiation pressure. The technique employs two frequency-offset laser fields to simultaneously create a stiff optical restoring force and a viscous optical damping force. We show how these forces may be used to optically trap a free mass without introducing thermal noise, and we demonstrate the technique experimentally with a 1 g mirror. The observed optical spring has an inferred Young's modulus of 1.2 TPa, 20% stiffer than diamond. The trap is intrinsically cold and reaches an effective temperature of 0.8 K, limited by technical noise in our apparatus

Müller-Ebhardt, H.

MPG.PuRe

arXiv:quant-ph/0612188v2  26 Jul 2007LIGO-P060025-00-RAn all-optical trap for a gram-scale mirrorThomas Corbitt,1 Yanbei Chen,2 Edith Innerhofer,1 Helge Mu¨ller-Ebhardt,3 David Ottaway,1Henning Rehbein,3 Daniel Sigg,4 Stanley Whitcomb,5 Christopher Wipf,1 and Nergis Mavalvala11LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA2Max-Planck-Institut fu¨r Gravitationsphysik, Am Mu¨hlenberg 1, 14476 Potsdam, Germany3Max-Planck-Institut fu¨r Gravitationsphysik (Albert Einstein Institute), Callinstraße 38, 30167 Hannover, Germany4LIGO Hanford Observatory, Route 10, Mile marker 2, Hanford, WA 99352, USA5LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA(Dated: January 21, 2008)We report on a stable optical trap suitable for a macroscopic mirror, wherein the dynamics of themirror are fully dominated by radiation pressure. The technique employs two frequency-offset laserfields to simultaneously create a stiff optical restoring force and a viscous optical damping force. Weshow how these forces may be used to optically trap a free mass without introducing thermal noise;and we demonstrate the technique experimentally with a 1 gram mirror. The observed optical springhas an inferred Young’s modulus of 1.2 TPa, 20% stiffer than diamond. The trap is intrinsicallycold and reaches an effective temperature of 0.8 K, limited by technical noise in our apparatus.The change in dynamics caused by radiation pres-sure effects has been explored in many mechanical sys-tems; its proposed applications include cooling towardthe ground state of nano- or micro-electromechanicalsystems (N/MEMS) [1, 2, 3, 4, 5, 6], enhancing thesensitivity of gravitational wave (GW) detectors [7, 8],and generation of ponderomotively squeezed light [9].Two types of radiation pressure effects are evident inthese systems: the optical restoring and viscous damp-ing forces, both of which are generated by detuned opticalcavities. Detuning a cavity to higher frequencies (blue-detuning) gives rise to a restoring force, known as an op-tical spring [10, 11, 12], as well as an anti-damping forcedue to the delay in the cavity response time. Conversely,detuning to lower frequencies (red-detuning) gives rise tooptical damping [13] along with an anti-restoring force.In N/MEMS, optical (anti-)restoring forces are typi-cally negligible in comparison to the stiff mechanical sus-pension. However, optical damping produces cooling ina red-detuned cavity, while anti-damping heats, or evenleads to instability in a blue-detuned cavity [1, 2, 3, 4,5, 14, 15]. In GW detectors, on the other hand, theoptical spring force may dominate, since the mechani-cal suspension of their mirrors is very soft. The typicaluse of the optical spring effect in these systems is to en-hance the sensitivity of the detector around the opticalspring resonance. To achieve a restoring force, the cavitymust be blue-detuned, and the coincident optical anti-damping force can both destabilize the cavity and giverise to parametric instabilities of the internal modes ofits mirrors [8, 16, 17]. In general, whenever the radia-tion pressure of a single optical field dominates both themechanical damping and restoring forces, the system isunstable due to the presence of a strong anti-damping oranti-restoring optical force. Hence, until now this regimehas been achieved only with the help of active feedbackcontrol to stabilize the dynamics [8, 16].Here we propose and demonstrate a technique that cir-LASERFIAOM EOMHWPPDHFrequency LengthPBS PBS 1 gram250 gramFIG. 1: (Color online) Simplified schematic of the experi-ment. About 3 W of λ0 = 1064 nm Nd:YAG laser lightpasses through a Faraday isolator (FI) before it is split intotwo paths by a half-waveplate (HWP) and polarizing beam-splitter (PBS) combination that allows control of the laserpower in each path. The carrier (C) field comprises most ofthe light incident on the suspended cavity. About 5% of thelight is frequency-shifted by one free spectral range (161.66MHz) using an acousto-optic modulator (AOM), and phasemodulated by an electro-optic modulator (EOM); this sub-carrier (SC) field can further be detuned from resonance tocreate a second optical spring. The two beams are recom-bined on a second PBS before being injected into the cavity,which is mounted on a seismic isolation platform in a vacuumchamber (denoted by the shaded box). A Pound-Drever-Hall(PDH) error signal derived from the SC light reflected fromthe cavity is used to lock it, with feedback to both the cavitylength as well as the laser frequency. By changing the fre-quency shift of the SC, the C can be shifted off resonance byarbitrarily large detunings. The low power SC beam (blue)passes through the EOM and AOM before being recombinedwith the high power C beam (red)..cumvents the optomechanical instability by using the ra-diation pressure of a second optical field, thus creating astable optical trap for a 1 gram mirror. This opens a newroute to mitigating parametric instabilities in GW de-tectors, and probing for quantum effects in macroscopicobjects.The experiment shown schematically in Fig. 1 was2performed to demonstrate the optical trapping scheme.The 250 gram input mirror of the L = 0.9 m long cavityis suspended as a pendulum with oscillation frequency of1 Hz for the longitudinal mode. The 1 gram end mirroris suspended by two optical fibers 300 µm in diameter,giving a natural frequency Ωm = 2 pi × 172 Hz for itsmechanical mode, with quality factor Qm = 3200. Onresonance, the intracavity power is enhanced relative tothe incoming power by a resonant gain factor 4/Ti ≈ 5×103, where Ti is the transmission of the input mirror, andthe resonant linewidth (HWHM) is γ = Ti c4L≈ 2 pi × 11kHz.If the resonance condition is exactly satisfied, the in-tracavity power depends quadratically on small changesin the length of the cavity. In this case the radiation pres-sure is only a second-order effect for the dynamics of thecavity. The constant (dc) radiation pressure is balancedthrough external forces; consequently, only fluctuationsof the radiation pressure are considered here. If the cavityis detuned from the resonance condition, the intracavitypower, and therefore the radiation pressure exerted onthe mirrors, becomes linearly dependent on the length ofthe cavity, analogous to a spring. The resulting springconstant is given in the frequency domain by [11, 16]K (Ω) = K0[1 + (δ/γ)2− (Ω/γ)2][1 + (δ/γ)2− (Ω/γ)2]2+ 4 (Ω/γ)2K0 =2cdPdL=128 pi I0 (δ/γ)T 2ic λ0[11 + (δ/γ)2], (1)where Ω is the frequency of the motion, and δ and I0 arethe detuning and input power of the laser, respectively.Note the dependence of K0 on the sign of δ. For δ > 0(in our convention), K > 0 corresponds to a restoringforce, while δ < 0 gives an anti-restoring force; we do notexplore this regime experimentally since it is always un-stable for our system (see Fig. 2). The light in the cavity(for δ ≪ γ) responds to mirror motion on a time scalegiven by γ−1. This delay has two effects. First, for highfrequency motion (Ω >∼ γ), the response of the cavity,and the corresponding radiation pressure, are reduced,and we see from Eq. (1) that K (Ω≫ γ) ≈ K0 (Ω/γ)−2.Second, the response of the cavity lags the motion, lead-ing to an additional force proportional to the velocity ofthe mirror motion — a viscous force with damping coef-ficient given by [11, 16]Γ (Ω) ≡2K (Ω)M γ[1 + (δ/γ)2− (Ω/γ)2] , (2)where M is the reduced mass of the two mirrors. Be-cause the cavity response lags the motion of the mirrors,a restoring spring constant implies a negative damping.Again we see that when both optical forces dominatetheir mechanical counterparts, the system must be un-stable.To stabilize the system we use two optical fields that re-spond on different time scales. One field should respondquickly, so that it makes a strong restoring force andonly a weak anti-damping force. The other field shouldrespond slowly, so that it creates a strong damping force,with only a minor anti-restoring force. This could beachieved with two cavities of differing bandwidths thatshare a common end mirror. However, it is simpler touse a single cavity and two fields with vastly differentdetunings. From Eqs. (1) and (2), taking Ω ≪ γ (validat the optical spring resonant frequency), we findΓK=2/ (M γ)1 + (δ/γ)2; (3)we see that an optical field with larger detuning has lessdamping per stiffness. The physical mechanism for thisis that at larger detunings, the optical field resonates lessstrongly than for smaller detunings, so the time scale forthe cavity response is shorter, leading to smaller opticaldamping. To create a stable system, we consider a car-rier field (C) with large detuning δC ≈ 3 γ that creates arestoring force, but also a small anti-damping force. Tocounteract the anti-damping, a strong damping force iscreated by injecting a subcarrier (SC) with small detun-ing δSC ≈ −0.5 γ. For properly chosen power levels ineach field, the resulting system is stable; we found a fac-tor of 20 higher power in the carrier to be suitable in thiscase. To illustrate the behavior of the system at all de-tunings, the various stability regions are shown in Fig. 2for this fixed power ratio. Point (d) in particular showsthat the system is stable for our chosen parameters.Next we highlight some notable features of this opticaltrapping technique that were demonstrated experimen-tally using the apparatus of Fig. 1.(i) Extreme rigidity: With no SC detuning andδC ≈ 0.5 γ, the 172 Hz mechanical resonance of the 1gram mirror oscillator was shifted as high as 5 kHz (curve(a) in Fig. 3), corresponding to an optical rigidity ofK = 2 × 106 N/m. To put this number into perspec-tive, consider replacing the optical mode with a rigidbeam with Young’s modulus E. The effective Young’smodulus of this mode with area A of the beam spot(1.5 mm2) and length L = 0.9 m of the cavity, is givenby E = K L/A = 1.2 TPa, stiffer than any known mate-rial (but also with very small breaking strength). Suchrigidity is required to operate the cavity without exter-nal control; ambient motion would otherwise disrupt thecavity resonance condition.(ii) Stabilization: Also shown in Fig. 3 are curvescorresponding to various C and SC detunings. In curves(b), (c) and (d), we detune the carrier by more than thecavity linewidth since the optical spring is less unstablefor large δC . With no SC detuning, the optomechanicalresonant frequency reaches Ωeff = 2 pi × 2178 Hz, shown3δC / γδ SC / γCold damping(a) (c)(b)(d)−5 −2.5 0 2.5 5−2−1012Statically unstableDynamically unstableStableAnti−stableFIG. 2: (Color online) Graphical representation of the to-tal optical rigidity due to both optical fields, as a functionof C and SC detuning, for fixed input power (power in theSC field is ∼ 1/20 the C power) and observation frequency(Ω = 2pi × 1 kHz). The shaded regions correspond to detun-ings where the total spring constant K and damping constantΓ are differently positive or negative. Specifically, “stable”corresponds to K > 0 ,Γ < 0, “anti-stable” to K < 0 ,Γ > 0,“statically unstable” to K < 0 ,Γ < 0, and “dynamicallyunstable” to K > 0 ,Γ > 0. The blue line denotes “colddamping” corresponding to δC < 0 and δSC = 0, i.e., theSC provides no optical force. The (logarithmically spaced)contours shown are scaled according to K: brighter regionshave larger K. The labels (a) – (d) refer to the measurementsshown in Fig. 3.in curve (c). Note that the optical spring is unstable, asevidenced by the phase increase of 180◦ about the reso-nance (corresponding to anti-damping). Next we detunethe subcarrier in the same direction as the carrier, shownin curve (b), which increases the resonant frequency andalso increases the anti-damping, demonstrated by thebroadening of the resonant peak. For these two cases,electronic servo control is used to keep the cavity locked.If the control system is disabled, the amplitude of the cav-ity field and mirror oscillations grow exponentially. Re-markably, when the subcarrier is detuned in the oppositedirection from the carrier, the optical spring resonancebecomes stable, shown in curve (d), allowing operationof the cavity without electronic feedback at frequenciesabove 30 Hz; we note the change in phase behavior andthe reduction of the resonant frequency. This shows howthe frequency and damping of the optical spring can beindependently controlled in the strong coupling regime.(iii) Optical cooling: The thermal excitation spec-trum of the mirror, given by SF = 4kBTΓm/M , is notchanged by the optical forces. It is informative to expressthis in terms of the optomechanical parameters Γeff , Ωeffand an effective temperature, Teff , such that the form ofthe equation is maintained. The effective temperature110100Magnitude (arb. units)(a)(b)(c)(d)103 104−180−90090180Ω/2pi (Hz)Phase (deg.) (a)(b)(c)(d)FIG. 3: The optical spring response for various power levelsand detunings of the carrier and subcarrier. Measured trans-fer functions of displacement per force are shown as points,while the solid lines are theoretical curves. The dashed lineshows the response of the system with no optical spring. Anunstable optical spring resonance with varying damping andresonant frequency is produced when (a) δC = 0.5 γ , δSC = 0;(b) δC = 3 γ , δSC = 0.5 γ; (c) δC = 3 γ , δSC = 0; and it is sta-bilized in (d) δC = 3 γ , δSC = −0.3 γ. Note that the dampingof the optical spring increases greatly as the optomechanicalresonance frequency increases, approaching Γeff ≈ Ωeff for thehighest frequency optical spring.thus is given byTeff = TΓmΓeff= TΩmΩeffQeffQm, (4)where Qi = Ωi/Γi (i = m, eff) is the quality factor ofthe oscillator. In the standard cold damping techniquelower Teff is achieved by decreasing Qeff via the viscousradiation pressure damping. The optical spring effectresults in further cooling by increasing the resonant fre-quency. The combination of both effects allows for muchcolder temperatures to be attained than with cold damp-ing alone. This is relevant to experiments hoping toobserve quantum effects in macroscopic objects, since itgreatly reduces the thermal occupation numberN =kBTeffh¯Ωeff, (5)both by decreasing the effective temperature, and in-creasing the resonant frequency.In the current experiment the displacement spectrumis dominated by laser frequency noise at Ωeff . We cannonetheless estimate the effective temperature of the op-tomechanical mode by measuring the displacement of themirror, and equating 12Kx2rms =12kBTeff , where xrms isthe RMS motion of the mirror. To determine xrms inour experiment, we measure the noise spectral density ofthe error signal from the cavity, calibrated by injecting41500 1600 1700 1800 1900 2000 2100 2200 2300?/2? (Hz)0.00.51.01.52.02.53.03.5????????????????????×10-15T = 0.8 KT = 12.2 KFIG. 4: (Color online) The measured noise spectral densityof the cavity length is shown for several configurations corre-sponding to different detunings. The lowest amplitude (ma-genta) curve corresponds to δC ≈ 3 and δSC ≈ −0.5. Theother (green and blue) curves are obtained by reducing δSCand increasingδC in order to keep Ωeff approximately con-stant, while varying Γeff . The spectrum is integrated between1500 and 2300 Hz to calculate the rms motion of the oscilla-tor mode, giving effective temperatures of 0.8, 3.8 and 12.2K. The limiting noise source here is not thermal noise, but infact frequency noise of the laser, suggesting that with reducedfrequency noise, even lower temperatures could be attained.a frequency modulation of known amplitude at 12 kHz.The displacement noise measured in this way is shownin Fig. 4. The lowest measured temperature of 0.8 Kcorresponds to a reduction in N by a factor of 2.5× 103.In conclusion, we have exhibited a scheme that usesboth the optical spring effect and optical damping fromtwo laser fields to create a stable optomechanical systemin which the dynamics are determined by radiation pres-sure alone. We experimentally demonstrated that thesystem is indeed stable, confirmed by deactivating theelectronic control system and permitting the cavity toevolve freely at the dynamically relevant frequencies. Webelieve this is a useful technique for manipulating the dy-namics of radiation pressure dominated systems, to quelltheir instabilities and examine their quantum behaviorfree from external control.We would like to thank our colleagues at the LIGOLaboratory and the MQM group, and also Steve Girvin;a casual discussion with Steve prompted some of theexperimental work explored in this manuscript. Weare also grateful to Mike Boyle for first promptingus to explore double optical springs, and to Ken-taro Somiya for helpful discussions and introduction tothe MQM group. We gratefully acknowledge supportfrom National Science Foundation grants PHY-0107417and PHY-0457264. Y.C.’s research was supported bythe Alexander von Humboldt Foundation’s Sofja Ko-valevskaja Award (funded by the German Federal Min-istry of Education and Research).[1] C. H. Metzger, K. Karrai, Nature 432, 1002 (2004).[2] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.Clerk, M. P. Blencowe, and K. C. Schwab, Nature443,193(2006).[3] S. Gigan, H. Bo¨hm, M. Paternostro, F. Blaser, G. Langer,J. Hertzberg, K. Schwab, D. Ba¨uerle, M. Aspelmeyer,and A. Zeilinger, Nature 444, 67 (2006).[4] D. Kleckner, D. Bouwmeester, Nature 444, 75 (2006).[5] O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A.Heidmann, Nature 444, 71 (2006).[6] A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, andT. J. Kippenberg, Phys. Rev. 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An All-Optical Trap for a Gram-Scale Mirror

Thomas Corbitt

Yanbei Chen

Edith Innerhofer

Helge Müller-Ebhardt

David Ottaway

Henning Rehbein

Daniel Sigg

Stanley Whitcomb

Christopher Wipf

Nergis Mavalvala

Crossref

We report on a stable optical trap suitable for a macroscopic mirror, wherein the dynamics of the mirror are fully dominated by radiation pressure. The technique employs two frequency-offset laser fields to simultaneously create a stiff optical restoring force and a viscous optical damping force. We show how these forces may be used to optically trap a free mass without introducing thermal noise, and we demonstrate the technique experimentally with a 1 g mirror. The observed optical spring has an inferred Young\u27s modulus of 1.2 TPa, 20% stiffer than diamond. The trap is intrinsically cold and reaches an effective temperature of 0.8 K, limited by technical noise in our apparatus. © 2007 The American Physical Society

Corbitt, Thomas

Chen, Yanbei

Innerhofer, Edith

Müller-Ebhardt, Helge

Ottaway, David

Rehbein, Henning

Sigg, Daniel

Whitcomb, Stanley

Wipf, Christopher

Mavalvala, Nergis

LSU Scholarly Repository (Louisiana State Univ.)

Max Planck Society eDoc Server

https://digital.library.adelaide.edu.au/dspace/bitstream/2440/48383/2/hdl_48383.pdf

An all-optical trap for a gram-scale mirror

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Adelaide Research & Scholarship

MPG.PuRe

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MPG.PuRe

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