Back-action evasion and squeezing of a mechanical resonator using a cavity detector
BBack-action evasion and squeezing of a mechanical resonator using a cavity detector
A. A. Clerk , F. Marquardt , and K. Jacobs Department of Physics, McGill University, Montr´eal, Qu´ebec, Canada, H3A 2T8 Department of Physics, Arnold-Sommerfeld-Center for Theoretical Physics,and Center for NanoScience, Ludwig-Maximilians-Universit¨at M¨unchen,Theresienstrasse 37, 80333 Munich, Germany and Department of Physics, University of Massachussets at Boston, Boston, MA 02125, USA (Dated: Feb. 13, 2008)We study the quantum measurement of a cantilever using a parametrically-coupled electromag-netic cavity which is driven at the two sidebands corresponding to the mechanical motion. Thisscheme, originally due to Braginsky et al. [V. Braginsky, Y. I. Vorontsov, and K. P. Thorne, Sci-ence , 547 (1980)], allows a back-action free measurement of one quadrature of the cantilever’smotion, and hence the possibility of generating a squeezed state. We present a complete quantumtheory of this system, and derive simple conditions on when the quantum limit on the added noisecan be surpassed. We also study the conditional dynamics of the measurement, and discuss howsuch a scheme (when coupled with feedback) can be used to generate and detect squeezed statesof the oscillator. Our results are relevant to experiments in optomechanics, and to experiments inquantum electromechanics employing stripline resonators coupled to mechanical resonators.
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I. INTRODUCTION
Considerable effort has been devoted recently to at-tempts at seeing quantum effects in micron to nanometrescale mechanical systems. Experiments coupling such os-cillators to mesoscopic electronic position detectors haveseen evidence of quantum back-action and back-actioncooling [1], and have demonstrated continuous positiondetection at a level near the fundamental limit placedby quantum backaction [2, 3, 4]. Complementary to thiswork, experiments using optomechanical systems (e.g. acantilever coupled to a optical cavity) have been ableto cool micromechanical resonators by several orders ofmagnitude, using either passive [5, 6, 7, 8] or active(i.e. feedback-based) approaches [9] .Despite these recent successes, seeing truly quantumbehaviour in a mechanical resonator remains a difficultchallenge. If one is only doing linear position detection,the quantum behaviour of an oscillator is almost per-fectly masked. Non-linear detector-oscillator couplingsallow one to probe quantum behaviour such as energyquantization [10, 11, 12, 13, 14]; however, generatingsuch couplings is generally not an easy task. Quan-tum behaviour could also be revealed by coupling theresonator to a qubit [15, 16, 17]; this too is challeng-ing, as it requires relatively large couplings and a highlyphase coherent qubit. Here, we consider an alternateroute to seeing quantum behaviour in a mechanical oscil-lator, one that requires no qubit and only a linear cou-pling to position. As was first suggested by Braginskyand co-workers [18, 19], by using an appropriately drivenelectromagnetic cavity which is parametrically coupledto a cantilever, one can make a measurement of just asingle quadrature of the cantilever’s motion. As a result,quantum mechanical back-action need not place a limiton the measurement accuracy, as the back-action affects only the unmeasured quadrature. One can then make(in principle) a perfect measurement of one quadratureof the oscillator’s motion. This is in itself useful, as it al-lows for the possibility of ultra-sensitive force detection[20, 21]. Perhaps even more interesting, one expects thatsuch a measurement can result in a quantum squeezedstate of the oscillator, where the uncertainty of the mea-sured quadrature drops below its zero point value.While the original proposal by Braginsky is quite old,there nonetheless does not exist a fully quantum theory ofthe noise and back-action of this scheme; moreover, thereexists no treatment of the measurement-induced squeez-ing. In this paper, we remedy this situation, and presenta fully quantum theory of measurement in this system.We calculate the full noise in the homodyned output sig-nal from the cavity (an experimentally measurable quan-tity), and derive simple but precise conditions that areneeded to beat the conventional quantum limit on theadded noise of a position detector [19, 20, 22, 23]. Us-ing a conditional measurement approach, we also discussthe conditions required to squeeze the mechanical res-onator, and demonstrate how feedback may be used tounambiguously detect this squeezing. Our results are es-pecially timely, given the recent experimental successes inrealizing cavity position detectors using both supercon-ducting stripline resonators [24] as well as optical cavi-ties [5, 6, 7, 8, 25]; our theory is applicable to both theseclasses of systems. Note that Ruskov et al. [26] recentlyanalyzed a somewhat related scheme involving strobo-scopic measurement of an oscillator with a quantum pointcontact. Unlike that scheme, the system analyzed hereshould be much easier to implement, being directly re-lated to existing experimental setups; our scheme alsohas the benefit of allowing significant squeezing withoutthe need to generate extremely fast pulses.The remainder of this paper is organized as follows. InSec. II we give a heuristic description of how one may a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b realize back-action free single-quadrature detection, in-troduce the Braginsky two-sideband scheme, and give asynopsis of our main findings. In Sec. III we provide thedetails of our calculations, and Sec. IV concludes. II. MODEL AND MAIN RESULTSA. Basic idea behind single quadrature detection
Consider a high-Q mechanical oscillator having fre-quency ω M and annihilation operator ˆ c . We will use X and Y to denote the cosine and sine quadratures of theoscillator’s motion. Using Schr¨odinger operators ˆ c andˆ c † , the operators associated with the quadratures are:ˆ X = 1 √ (cid:0) ˆ ce iω M t + ˆ c † e − iω M t (cid:1) (1a)ˆ Y = − i √ (cid:0) ˆ ce iω M t − ˆ c † e − iω M t (cid:1) (1b)The Heisenberg-picture position operator ˆ x ( t ) is thengiven by the Heisenberg picture operators ˆ X ( t ) and ˆ Y ( t )via:ˆ x ( t ) ≡ √ x zpt (cid:16) ˆ X ( t ) · cos ω M t + ˆ Y ( t ) sin ω M t (cid:17) (2)as expected. Note that ˆ X and ˆ Y are canonically conju-gate: (cid:104) ˆ X, ˆ Y (cid:105) = i (3)Also note that the definition of the quadrature operatorsrelies on having an external clock in the system whichdefines the zero of time.In general, ˆ X ( t ) and ˆ Y ( t ) will vary slowly in time (incomparison to ω M ) due to the external forces acting onthe oscillator. Our goal will be to make a weak, con-tinuous measurement of only ˆ X ( t ), using the usual kindof setup where the position of the oscillator is linearlycoupled to a detector. We will use a detector - oscillatorcoupling Hamiltonian of the form:ˆ H int = − A ˆ x ˆ F (4)where ˆ F is some detector operator. It represents theforce exerted by the detector on the oscillator; in thecavity-position detector we will consider, ˆ F will be thenumber of photons in an electromagnetic cavity. Ideally,if we only measure ˆ X , the back-action of the measure-ment will only affect the unmeasured quadrature ˆ Y , andwill not affect the evolution of ˆ X at later times. The hopethus exists of being able to make a back-action free mea-surement, one which is not subject to the usual standardquantum limit [20, 22, 23].As is discussed extensively in Refs. 20 and 21, singlequadrature detection with the interaction Hamiltonian inEq. (4) can be accomplished by simply modulating the FIG. 1: Schematic picture of the setup studied in the text.A cavity is driven by an input beam that is amplitude-modulated at the mechanical frequency ω M of the movableend-mirror. The radiation pressure force, as well as the cav-ity and mechanical frequencies and decay rates are indicated. coupling strength A at the oscillator frequency. Setting A = A ( t ) = 2( ˜ A/x zpt ) cos( ω M t ), ˆ H int becomes:ˆ H int = −√ A ˆ F · (cid:104) ˆ X (1 + cos(2 ω M t )) + ˆ Y sin(2 ω M t ) (cid:105) (5)In a time-averaged sense, we see the detector is only cou-pled to the X quadrature; we thus might expect that the(time-averaged) output of the detector will tell us onlyabout X . In principle, this in itself does not imply a lackof back-action: via the coupling to ˆ Y , noise in ˆ F could af-fect the dynamics of ˆ X . To prevent this, we need the fur-ther requirement that the detector force has no frequencycomponents near ± ω M . In this case, the effective back-action force ˆ F sin(2 ω M t ) will have no Fourier weight inthe narrow-bandwidth around zero frequency to which ˆ X is sensitive, and it will not affect ˆ X . Note that Ruskovet al. [26] recently considered a linear position detectionscheme where the effective coupling constant is harmon-ically modulated; however, their scheme does not satisfythe second requirement above of having a narrow-bandback-action force. B. Model
We now consider a specific and experimentally-realizable system which can realize the above ideas; thissystem was first proposed by Braginsky [18, 19]. Asshown schematically in Fig. 1, the setup consists of ahigh-Q mechanical oscillator which is parametrically cou-pled with strength A to a driven electromagnetic cavity:ˆ H = (cid:126) ( ω R − A ˆ x ) (cid:2) ˆ a † a − (cid:104) ˆ a † a (cid:105) (cid:3) + ˆ H M + ˆ H drive + ˆ H κ + ˆ H γ (6)where ω R is the cavity resonance frequency, ˆ H M = (cid:126) ω M ˆ c † ˆ c is the mechanical oscillator Hamiltonian, ˆ H drive describes the cavity drive, ˆ H κ describes the cavity damp-ing, and ˆ H γ is the mechanical damping. Note this samesystem was recently shown (in a similar parameter regimeto what we will require) to allow back-action cooling tothe ground state [27].Assuming a one-sided cavity, standard input-outputtheory [28, 29] yields the Heisenberg equation of motion:˙ˆ a = (cid:16) − iω R − κ (cid:17) ˆ a − √ κ ˆ b in ( t ) . (7)where κ is the cavity damping, and ˆ b in describes both thedrive applied to the cavity as well as the noise (quantumand thermal) entering the vacuum port.To implement back-action evasion in the cavity system,we will consider the case where ω R (cid:29) ω M , and take theresolved-sideband or “good cavity” limit, where ω M (cid:29) κ .We will also take an amplitude-modulated cavity drive ofthe form: (cid:104) ˆ b in ( t ) (cid:105) ≡ ¯ b in ( t ) = ¯ b LO ω M t ) e − iω R t (8)The same resolved sideband limit is required to achieveground state cooling [27]; all that is different from thesetup here is the nature of the drive. Here one drives thecavity equally at both sidebands associated with the os-cillator motion, while in the cooling case, one only drivesthe red-detuned sideband.To proceed, we may write the cavity annihilation oper-ator ˆ a as the sum of a classical piece ¯ a ( t ) and a quantumpiece ˆ d : ˆ a ( t ) = ¯ a ( t ) + ˆ d ( t ) (9)¯ a ( t ) is determined solely by the the response of the cavityto the (classical) external drive ¯ b in ( t ). In the long-timelimit, Eq. (8) yields:¯ a ( t ) = ¯ a max cos ( ω M t + δ ) e − iω R t (10)with ¯ a max = ¯ b LO (cid:114) κ ω M + κ (11a) δ = arctan ( κ/ω M ) (11b)The phase δ plays no role except to set the definitionsof the two quadratures X and Y ; thus, without loss ofgenerality, we will set it to zero. We will also be interestedin a drive large enough that ¯ a max (cid:29) a , ˆ d , the quantum part of the cavity an-nihilation operator, is influenced by both the mechanicaloscillator and quantum noise associated with the cavitydissipation. Making use of the solution for ¯ a and the con-ditions ω R (cid:29) ω M (cid:29) κ , and keeping only terms which areat least order ¯ a , the term in the total system Hamiltoniancoupling the oscillator to the cavity takes an analogousform to Eq. (5) with:˜ A = 12 ( Ax zpt ) ¯ a max (12a)ˆ F = e iω R t ˆ d + e − iω R t ˆ d † (12b)Thus, the chosen cavity drive gives us the requiredharmonically-modulated coupling constant: in a time-averaged sense, the cavity is only coupled to the X oscil-lator quadrature. Further, the second condition outlined in Sec. II A is also satisfied: because κ (cid:28) ω M , ˆ F has noappreciable noise power at frequencies near ± ω M . Assuch, we expect no back-action heating of the ˆ X quadra-ture in the resolved-sideband limit κ/ω M →
0. We will ofcourse consider the effect of a non-zero but small κ/ω M in what follows. Note that electromechanical realizationsof the system presented here, which use superconductingstripline resonators, can easily achieve the resolved side-band limit ω M (cid:29) κ (e.g. the recent experiment of Ref. 24achieved ω M /κ ≈ ω M /κ ≈
20 [25].
C. Back-action
Working in an interaction picture, one can easily de-rive Heisenberg equations of motion for the system, andsolve these in the Fourier domain (c.f. Eqs. (43)). Asexpected, one finds that in the ideal good cavity limit( κ/ω M → X quadrature is completelyunaffected by the coupling to the cavity, while the un-measured Y quadrature experiences an extra back-actionforce due to the cavity. For finite κ/ω M , there is somesmall additional back-action heating of the X quadra-ture. The noise spectral density of the quadrature fluc-tuations are given by: S X ( ω ) ≡ (cid:90) ∞−∞ dte iωt (cid:68) { ˆ X ( t ) , ˆ X (0) } (cid:69) = γ/ ω + ( γ/ [1 + 2 ( n eq + n bad )] (13a) S Y ( ω ) = 12 (cid:90) ∞−∞ dte iωt (cid:68) { ˆ Y ( t ) , ˆ Y (0) } (cid:69) = γ/ ω + ( γ/ [1 + 2 ( n eq + n BA + n bad )](13b)where n eq = (cid:18) exp (cid:20) (cid:126) ω M k B T (cid:21) − (cid:19) − (14)is the number of thermal quanta in the oscillator. n BA parameterizes the back-action heating of the Y quadra-ture as an effective increase in n eq ; in the relevant limit γ (cid:28) κ one has: n BA = 8 ˜ A κγ = 2 ( Ax zpt ) κγ (¯ a max ) (15)We have assumed here that the there is no thermal noisein the cavity drive: it is shot noise-limited.Finally, n bad parameterizes the spurious back-actionheating of X which occurs when one deviates from thegood-cavity limit; to leading order in κ/ω M , it is simplygiven by: n bad = n BA (cid:18) κω M (cid:19) (16)Note that there is no back-action damping of eitherquadrature (see discussion following Eqs. (43)). D. Output Spectrum and Beating the SQL
We assume that a homodyne measurement is madeof the light leaving the cavity. Using the solution to theHeisenberg equations of motion (c.f. Eqs. (43)) and stan-dard input-output theory, one can easily find the noisespectral density of the homodyne current I ( t ). The in-formation about ˆ X ( t ) will be contained in a bandwidth ∼ γ (cid:28) κ around zero frequency. Thus, focusing on fre-quencies ω (cid:28) κ , we have simply: S I ( ω ) = G (cid:20) S X ( ω ) + κ
32 ˜ A S (cid:21) (17)Here, G is a gain coefficient proportional to the homo-dyne local oscillator amplitude, and S represents addednoise in the measurement coming from both the cavitydrive and in the homodyne detection. If both are shotnoise limited, we simply have S = 1. We can refer thisnoise back to the oscillator by simply dividing out thefactor G : the result is the measured X quadrature fluc-tuations: S X, meas ( ω ) ≡ S I ( ω ) G = S X ( ω ) + κ
32 ˜ A S (18)Now, note that in the good cavity limit the spurious heat-ing of X described by n bad vanishes. Thus, in this limit,the added noise term (second term in Eq. (18)) can bemade arbitrarily small by increasing the intensity of thecavity drive beam (and hence ˜ A ), without any result-ing back-action heating of the measured X quadrature.Thus, in the good-cavity limit, there is no back-action im-posed limit on how small we can make the added noise ofthe measurement (referred back to the oscillator). In con-trast, for small but non-zero κ/ω M , one needs to worryabout the small residual back-action described by n bad ;one can still nonetheless beat the standard quantum limitin this case, as we now show.To compare against the standard quantum limit, con-sider S X, meas (0): S X, meas (0) = 2 γ (1 + 2 n eq + 2 n bad ) + κ
32 ˜ A S ≡ γ (1 + 2 n eq + 2 n add ) (19)In the last line, we have represented both the residualback-action n bad and the added noise of the measurementas an effective increase in the number of oscillator quanta A dded no i s e ( add ) BA ! / " = 0, S = 1 ! / " = 0.5, S = 1 ! / " = 0, S = 10 ! / " = 0.5, S = 10 FIG. 2: Plot of added noise in the single quadrature measure-ment (measured as a number of quanta, n add , c.f. Eq. (20))versus the strength of the measurement (measured in terms ofthe back-action heating of the Y quadrature, n BA , c.f. Eq. 15).Different curves correspond to different values of κ/ω M and S , the noise associated with the homodyne measurement; S = 1 corresponds to a shot-noise limited measurement. Thestandard quantum limit of n add = 0 . by an amount n add . The standard quantum limit (whichapplies when both quadratures are measured) yields thecondition n add ≥ / n add = n bad + κγ
128 ˜ A S = n BA (cid:18) κω M (cid:19) + 116 n BA S (20)Thus, if we are in the ideal good-cavity limit ( κ/ω M → n add requires a coupling strong enough that n BA ≥ /
8: the Y quadrature fluctuations must beheated up by at least an eighth of an oscillator quantum.In the more general case where κ/ω M is finite, onecannot increase the coupling indefinitely, as there is back-action on ˆ X . One finds that for an optimized couplingof: ˜ n BA = ω M κ (cid:112) S (21)the minimum added noise at resonance is given by: n add (cid:12)(cid:12)(cid:12) min = κ ω M (cid:114) S κ/ω M , one can make n add smaller than the standard quantum limit value (seeFig. 2). E. Conditional Squeezing
Given that the double-sideband scheme described herecan allow for a near perfect measurement of the os-cillator X quadrature, one would expect it could leadto a squeezed oscillator state, where the uncertainty inˆ X drops below the zero point value of 1 /
2. However,Eq. (13a) indicates that in the good cavity limit, the fluc-tuations of ˆ X are completely unaffected by the couplingto the cavity detector. To resolve this seeming contra-diction, one must consider the conditional aspects of themeasurement: what is the state of the resonator in aparticular run of the experiment? In any given run ofthe experiment, the oscillator will indeed be squeezed.However, the mean value of ˆ X will have some non-zerovalue which is correlated with the noise in the outputsignal. Once one averages over many realizations of theexperiment, this random motion of (cid:104) ˆ X (cid:105) appears as extranoise, and masks the squeezing, resulting in the resultof Eq. (13a). We make these statements precise in whatfollows.A rigorous description of the conditional evolution ofthe oscillator in the setup considered here can be devel-oped in analogy to Ref. [30], which considered ordinarylinear position detection using a cavity. For simplicity,we focus on the good cavity limit, where κ/ω M → k , a measure of the rate atwhich the measurement extracts information, as:˜ k = η
32 ˜ A κ = η (4 γn BA ) (23)where n BA represents as before the back-action heatingof the Y quadrature, and η = S ≤ η = 1 correspondsto being quantum limited). One has ˜ k = 1 /τ meas , where τ meas is the minimum time required to resolve a differencein (cid:104) X (cid:105) equal to the zero point rms value from the outputof the detector; as we are interested in weak measure-ments, we expect ˜ k/ω M (cid:28)
1. Note that ˜ k = 8 ηk , where k is the usual definition of the strength of the measure-ment [31]. The scaled homodyne output signal may thenbe written [31]: I ( t ) = (cid:112) ˜ k (cid:104) ˆ X ( t ) (cid:105) + ξ ( t ) (24)where ξ ( t ) is white Gaussian noise. In a given run of theexperiment, ξ ( t ) will be correlated with the state of theoscillator at times later than t .In exact analogy to Ref. 30, a simple description ofthe conditional density matrix is possible in the limitwhere κ (cid:29) ˜ A . In this limit, the oscillator density ma-trix is Gaussian, being fully determined by its means¯ X = (cid:104) ˆ X (cid:105) , ¯ Y = (cid:104) ˆ Y (cid:105) and its second moments V X = (cid:104)(cid:104) ˆ X (cid:105)(cid:105) , V Y = (cid:104)(cid:104) ˆ Y (cid:105)(cid:105) and C = (cid:104)(cid:104){ ˆ X, ˆ Y } / (cid:105)(cid:105) . In theinteraction picture (i.e. rotating frame at the oscillatorfrequency), the equations for the means (the estimates) are ˙¯ X = − γ X + (cid:112) ˜ kV X ξ (25a)˙¯ Y = − γ Y + (cid:112) ˜ kCξ (25b)and for the covariances are˙ V X = − ˜ kV X − γ ( V X − ˜ T eq ) (25c)˙ V Y = − ˜ kC + ˜ k/ (4 η ) − γ ( V Y − ˜ T eq ) (25d)˙ C = − γC − ˜ kV X C (25e)where: ˜ T eq = 12 + n eq (26)We stress that these equations are almost identical tothe standard equations for conditional linear positiondetection [26, 30], with the important exception thatterms corresponding to the bare oscillator Hamiltonianare missing. In a sense, the scheme presented here effec-tively transforms away the oscillator Hamiltonian.To find the amount of squeezing in a particular run ofthe experiment, we simply find the stationary variancesfor the oscillator’s Gaussian state. We have: V Y = 12 + n eq + n BA (27a) V X = (cid:112) n eq ) ( ηn BA ) + 1 / − /
24 ( ηn BA ) (27b) C = 0 (27c)Note first that the result for V Y is in complete agree-ment with the unconditional result of Eq. (13b): themeasurement back-action heats the Y quadrature by anamount corresponding to n BA quanta. In contrast, wefind that unlike the unconditional result of Eq. (13a),the measurement causes V X to decrease below its zero-coupling value: it is a monotonically decreasing functionof n BA (see Fig. 3). This is the expected measurement-induced squeezing. Of particular interest is the minimumcoupling strength needed to reduce V X to its zero-pointvalue: n BA = n eq η (28)In other words, lowering the X quadrature uncertaintyfrom a thermal value of (1 / n eq ) to the ground statevalue of (1 /
2) requires that we at least increase the Y -quadrature uncertainty by the same amount. This mini-mum amount is only achieved for a quantum-limited de-tector η = 1.The equation describing the fluctuations of the meanquadrature amplitudes ¯ X, ¯ Y can also be easily solved.Assuming that ¯ X = ¯ Y = 0 at the initial time, one al-ways has (cid:104) ¯ X ( t ) (cid:105) = (cid:104) ¯ Y ( t ) (cid:105) = 0, where the average hereis over many runs of the experiment. In the stationarystate (i.e. once the variances V X , V Y and C have attained quad r a t u r e v a r i an c e ( ) BA FIG. 3: Plot of the conditional
X-quadrature variance V X (c.f. Eq. (27b)) as a function of the measurement strength(parametrized in terms of the back-action heating of the Y quadrature, n BA , c.f. Eq. (15)); one clearly sees that the X -quadrature can be squeezed. Different curves correspond todifferent values of the bath temperature (parameterized by n eq , c.f. Eq. (14)) and measurement efficiency η . The solidred curve corresponds to n eq = 0 and η = 1; the dashed blackcurve to n eq = 1 and η = 1; the dashed-dot blue curve to n eq = 0, η = 0 .
1. The horizontal dashed line corresponds tothe ground state value of the variance, V X = 0 . their stationary value), one finds ¯ Y ( t ) = 0 with no fluc-tuations. ¯ X ( t ) continues to fluctuate, with an autocor-relation function: (cid:104) ¯ X ( t ) ¯ X (0) (cid:105) = (cid:32) ˜ kγ (cid:33) V X e − γ | t | / (29)Again the average here is over many runs of the experi-ment.We may now combine the results of Eq. (29), andEq. (25c) to find V X, tot , the total (unconditional) X vari-ance. One finds the simple result (valid in the stationaryregime): V X, tot ≡ V X + (cid:104) ¯ X (cid:105) = V X + ˜ kγ V X = 12 + n eq (30)This shows that, as expected, averaging the results of theconditional theory over many measurement runs repro-duces the result of the unconditional theory (i.e. the fluc-tuations of the measured X quadrature are completelyunaffected by the measurement). F. Feedback for true squeezing
In the previous section we saw how the state of the res-onator, once conditioned on the measurement record, is squeezed. We can use feedback control to turn this con-ditional squeezing into “real” squeezing of the resonator,where the full, unconditional oscillator variance V X, tot (c.f Eq. (30)) drops below the zero-point value. This isaccomplished by applying a time-dependent force to theresonator which is proportional to ¯ X ( t ), the measuredvalue of the X quadrature. Such a force can be used tosuppress the fluctuations in the mean value of X , andin the limit of strong feedback, can remove them com-pletely. The only fluctuations that remain are quantifiedby the conditional variances, which are squeezed. Notethat a similar approach was considered in Ref. [26].More precisely, if one makes the measurement at rate ˜ k described above (c.f. Eq. (23)), and applies the feedbackforce F ( t ) = αγ ¯ X sin ω M t in the laboratory frame, theresult is an effective damping of the X quadrature at arate αγ/
2. Calculating the fluctuations of the X quadra-ture under this feedback (the details of which are givenin the next section), we find that the total unconditioned X quadrature variance reaches a stationary state: V fb X, tot = (1 + α + ˜ kγ V X ) V X α (31)Here, V X is the conditioned variance given in Eq. (27b).Note that when α →
0, we again get the result ofEq. (30): the unconditioned X quadrature variance isnot affected by the measurement. In contrast, in thelimit of large α , one has V fb X, tot → V X . Thus, as claimedabove, in the limit of strong feedback, the total fluctua-tions of the X quadrature are reduced to the conditionalvariance; it may thus be squeezed.It is also important to ask how this squeezing will man-ifest itself in the measurement signal. Calculating S fb I ( ω ),the spectrum of the homodyne current in the presence offeedback, and referring it back to the X quadrature, wefind: S fb X, meas ( ω ) ≡ S fb I ( ω )˜ k = 1˜ k + ( n eq + 1 /
2) + αV X ω + (cid:2) γ (1 + α ) (cid:3) (32)In the absence of feedback ( α → / ˜ k ) plus the measurement-independent X -quadraturefluctuation spectrum S X ( ω ); this is in complete agree-ment with the unconditional theory (c.f. Eq. (19)). Whenfeedback is turned on, the second Lorentzian term inthe spectrum is modified; this corresponds to the addeddamping and noise caused by the feedback. In the limit ofstrong feedback, where α → ∞ , we find somewhat sur-prisingly that all signatures of the oscillator disappear: S fb X, meas ( ω ) → / ˜ k . Thus, in the limit of strong feedback,while the fluctuations in the X quadrature are V X , theydo not appear at all in the output signal! To understandwhy this is, note that in driving the resonator with aforce proportional ¯ X , we are driving it with a signal thatis correlated with the noise in the output signal. Thus,feedback leads to new correlations between the fluctua-tions of X and the output noise. In the limit of strongfeedback, one finds that the fluctuations of X have avariance V X , but are perfectly negatively correlated withthe output noise. The result is that the output noise iscompletely independent of V X .The above effect, in which the fluctuations of X vanishin the output signal, may be regarded as an example of noise squashing [32, 33, 34]. This is when one uses feed-back specifically to reduce the fluctuations in the out-put signal, rather than to reduce the fluctuations of thesystem being measured. This is possible only becausethe feedback uses the output signal, and thus correlatesthe system’s fluctuations with the output fluctuations.Strictly speaking, squashing refers only to the outputsignal that is part of the feedback loop (the so-called “in-loop signal”), not to the actual system being measured.While the existence of squashing in no way invalidatesthe real squeezing produced by the feedback, it does makeit more difficult to observe this squeezing at the detec- tor output. Further, any experimental results may besubject to the accusation that the feedback protocol mayhave been incorrectly designed to produce only squash-ing, with the result that squeezing of the resonator couldnot be inferred from the spectrum of the output.A solution to this problem is to make a second mea-surement of the mechanical resonator’s X quadrature(e.g. by using a second cavity coupled to the resonator).The measurement signal from this second measurement, I ( t ), is not subject to squashing because it is not part ofthe feedback loop. As a result, the measurement noise in I ( t ) is completely uncorrelated with the feedback signal.Since the second measurement is also a QND measure-ment of the X quadrature, it does not affect the resultsfor V fb X, tot or S fb X, meas ( ω ) derived above. If the rate of thesecond measurement is ˜ λ , then the spectrum of its output(again, referred back to the oscillator) is S fb X, meas , ( ω ) ≡ S fb I ( ω )˜ λ = 1˜ λ + 4 γ ( n eq + 1 / − A (2 ω/γ ) + (1 + α ) + A (2 ω/γ ) + (cid:16) kV X /γ (cid:17) (33)where: A = α (2 n eq + 1)(1 + 2˜ kV X /γ + α ) − αV X α (2 + α ) − n eq + 1)˜ k (34)The first term in the spectrum above represents theadded noise of the measurement (e.g. shot noise), whilethe terms in square brackets are a direct measure ofthe oscillator’s X quadrature fluctuations. We seethat with feedback, these are described by the sum oftwo Lorentzians. The integral of the area under thesepeaks directly yields V fb X, tot , the total (unconditional) X quadrature variance in the presence of feedback: (cid:90) dω π (cid:18) S fb X, meas , ( ω ) − λ (cid:19) = V fb X, tot (35)It thus follows from Eq.(31) that in the strong feedbacklimit ( α → ∞ ), the area under the resonant peak in theoutput spectrum directly yields V X , and hence a directmeasure of squeezing. Note that in this limit only thesecond Lorentzian term in S fb X, meas , ( ω ) survives, as onehas: lim α →∞ A = (2 n eq + 1) − V X = V X (cid:16) kV X /γ (cid:17) (36)where we have made use of Eq. (30). Thus, for strongfeedback, the squeezing of the oscillator can now be un-ambiguously detected in the output signal of the secondmeasurement: one obtains a simple Lorentzian resonancewhose area is simply V X . We remind the reader that in the same limit, the spectrum of the first measurementshows no signature of the oscillator. Note that in prac-tice, the limit of strong feedback is already achieved when α (cid:29) max(1 , ˜ k/γ = 4 n BA ). III. DETAILS OF CALCULATIONSA. Spectrum of the Detector Output
1. Equations of motion
The Heisenberg equations of motion (in the rotatingframe) follow directly from H and the dissipative termsin the total Hamiltonian:˙ˆ d = − κ d − √ κ ˆ ξ ( t ) e iω R t − i ˜ A (cid:2) c (cid:0) e − iω M t (cid:1) + h.c. (cid:3) = − κ d − √ κ ˆ ξ ( t ) e iω R t − i √ A (cid:104) ˆ X (1 + cos(2 ω M t )) + ˆ Y sin(2 ω M t ) (cid:105) (37a)˙ˆ c = − γ c − √ γ ˆ η ( t ) e iω M t − i ˜ A (cid:0) e iω M t (cid:1) (cid:16) ˆ d + ˆ d † (cid:17) (37b)Here, ˆ ξ describes noise in the cavity input operator ˆ b in .In the limit where there is only quantum noise (i.e. shotnoise) in the cavity drive, we have: (cid:104) ˆ ξ † ( t ) · ˆ ξ ( t (cid:48) ) (cid:105) = 0 (38a) (cid:104) ˆ ξ ( t ) · ˆ ξ † ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) (38b)In contrast, ˆ η describes equilibrium noise due to theintrinsic damping of the mechanical oscillator. One has: (cid:104) ˆ η † ( t ) · ˆ η ( t (cid:48) ) (cid:105) = n eq δ ( t − t (cid:48) ) (39a) (cid:104) ˆ η ( t ) · ˆ η † ( t (cid:48) ) (cid:105) = ( n eq + 1) δ ( t − t (cid:48) ) (39b)where n eq is a Bose-Einstein occupation factor evaluatedat energy (cid:126) ω M and temperature T bath .The equations of motion are easily solved by first writ-ing them in terms of the quadrature operators ˆ X and ˆ Y ,and then Fourier transforming. To present these solu-tions, we first introduce the cavity and mechanical oscil-lator susceptibilities as: χ R ( ω ) = 1 − iω + κ/ χ M ( ω ) = 1 − iω + γ/ , (40b)and define the back-action force ˆ f BA viaˆ f BA ( ω ) = − ˜ A √ κχ R ( ω ) (cid:16) ˆ ξ ( ω + ω R ) + ˆ ξ ( ω − ω R ) (cid:17) . (41)Note that while ˆ ξ describes white noise, the cavity sus-ceptibility χ R ( ω ) ensures that ˆ f BA ( ω ) is only significantaround a narrow bandwidth centered about zero fre-quency. Note also that we define Fourier transformedoperators via: ˆ A ( ω ) ≡ (cid:90) ∞−∞ dt ˆ A ( t ) e iωt (42a)ˆ A † ( ω ) ≡ (cid:90) ∞−∞ dt (cid:104) ˆ A ( t ) (cid:105) † e iωt (42b)As such, one has (cid:104) ˆ A ( ω ) (cid:105) † = ˆ A † ( − ω ).The solutions of the Fourier-transformed quadratureoperators then read:ˆ X ( ω ) = χ M ( ω ) (cid:34) − (cid:114) γ (cid:0) ˆ η ( ω + ω M ) + ˆ η † ( ω − ω M ) (cid:1) + ˆ f BA ( ω + 2 ω M ) − ˆ f BA ( ω − ω M )2 i (cid:35) (43a)ˆ Y ( ω ) = χ M ( ω ) (cid:34) i (cid:114) γ (cid:0) ˆ η ( ω + ω M ) − ˆ η † ( ω − ω M ) (cid:1) − ˆ f BA ( ω ) − ˆ f BA ( ω + 2 ω M ) + ˆ f BA ( ω − ω M )2 (cid:35) (43b) Note from Eqs. (43) that there is no back-action damp-ing of either quadrature, even when one deviates fromthe good cavity limit by having κ/ω M >
0. This is easyto understand on a purely classical level. Note first thatthat it is only the cosine quadrature (i.e. ˆ d + ˆ d † ) of thecavity which couples to the mechanical resonator. As thecavity is itself a harmonic oscillator, this means that onlythe cavity sine quadrature (i.e. ˆ d − ˆ d † ) will be affectedby the resonator motion. As the cavity cosine quadra-ture provides the back-action force on the resonator (c.f.Eq. (12b)), it thus follows that the back-action force is completely independent of both quadratures of the me-chanical resonator’s motion. There is thus no back-actiondamping, as such damping requires a back-action forcewhich responds (with some time-lag) to the motion of theoscillator.Equations (13) for the noise spectra of ˆ X and ˆ Y atfrequencies ω (cid:28) κ now follow directly from Eqs. (43)and Eqs. (38),(39) which determine the noises ˆ ξ and ˆ η .
2. Output Spectrum and Beating the SQL
Standard input-output theory [28, 29] yields the fol-lowing relation between ˆ b out , the field leaving the cavity,and ˆ b in , the field entering the cavity:ˆ b out ( t ) = ˆ b in ( t ) + √ κ ˆ a ( t ) (44)In our case of a one-sided cavity, this relation becomes inthe lab (i.e. non-rotating) frame: b out ( ω ) = ¯ b out ( ω ) + (cid:20) − i ( ω − ω R ) − κ/ − i ( ω − ω R ) + κ/ (cid:21) ˆ ξ ( ω ) − i ˜ A √ κχ R ( ω − ω R ) · ˆ X ( ω − ω R ) (45)The first term on the RHS simply represents the out-put field from the cavity in the absence of the mechan-ical oscillator and any fluctuations. It will yield sharppeaks at the two sidebands associated with the drive, ω = ω R ± ω M . The second term on the RHS of Eq. (45)represents the reflected noise of the incident cavity drive.This noise will play the role of the “intrinsic output noise”or “measurement imprecision” of this detector.Finally, the last term on the RHS of Eq. (45) is theamplified signal: it is simply the amplified quadrature X of the oscillator. We see that the dynamics of ˆ X willresult in a signal of bandwidth ∼ γ centered at the cav-ity resonance frequency. This can be detected by makinga homodyne measurement of the signal leaving the cav-ity. Using a local-oscillator amplitude b LO ( t ) = iBe − iω R t with B real, and defining the homodyne current as:ˆ I ( t ) = (cid:16) b ∗ LO ( t ) + ˆ b † out ( t ) (cid:17) (cid:16) b LO ( t ) + ˆ b out ( t ) (cid:17) (46)one finds that the fluctuating part of I is given infrequency-space by:ˆ I ( ω ) = − B (cid:104) √ A √ κχ R ( ω ) ˆ X ( ω ) + (47) i iω + κ/ iω − κ/ (cid:16) ˆ ξ ( ω R + ω ) − ˆ ξ † ( − ω R + ω ) (cid:17) (cid:105) The signal associated with the oscillator will be in a band-width ∼ γ (cid:28) κ : for these frequencies, the above expres-sion simplifies to:ˆ I ( ω ) = − B (cid:104) A (cid:112) κ/ X ( ω ) + (48) − i (cid:16) ˆ ξ ( ω R + ω ) − ˆ ξ † ( − ω R + ω ) (cid:17) (cid:105) Using this equation along with Eqs. (43a), (38) and(39), it is straightforward to obtain the result for thehomodyne spectrum S I ( ω ) given in Eq. (17). B. Conditional Evolution
To derive the stochastic master equation describing theconditional evolution of the resonator under the doublesideband measurement scheme, (that is, the evolutiongiven the continuous stream of information obtained bythe observer), one uses a procedure that is essentiallyidentical to that given in Ref. [30]. Once we have movedinto the interaction picture (in which the quadratures areQND observables), the displacement picture [35] (that is,separated ˆ a into ¯ a and ˆ d as per Eq. (9)), and made therotating-wave approximation, the Hamiltonian for thecombined cavity and resonator system is H = −√ A ( ˆ d + ˆ d † ) X. (49)We now perform homodyne detection of output from the(one-sided) cavity, and as a result the evolution of thesystem is given by the quantum optical stochastic masterequation [35, 36] dσ = − i (cid:126) [ H, σ ] dt + κ D [ ˆ d ] σdt + √ ηκ H [ − i ˆ d ] σdW, (50)where σ is the joint density matrix of the two systemsas before η is the detection efficiency, and κ is the cavitydecay rate. The superoperators D and H are given by2 D [ˆ c ] σ = 2ˆ cσ c ˆ c † − ˆ c † ˆ cσ − σ ˆ c † ˆ c, (51) H [ˆ c ] σ = ˆ cσ + σ ˆ c † − Tr[ˆ cσ + σ ˆ c † ] σ, (52)for an arbitrary operator ˆ c .We now wish to obtain an equation for the evolutionof the resonator alone. This is possible so long as thecavity decay rate is fast compared to the timescale of thecavity-resonator interaction. That is,˜ A (cid:112) (cid:104) X (cid:105) κ ∼ γκ ≡ (cid:15) (cid:28) , (53) This means that the light ouput from the cavity spendssufficiently little time in the cavity that it continuallyprovides up-to-the-minute information about the oscilla-tor. With this large damping rate, the fluctuations of thelight in the cavity about the average value ¯ a are small,and we can thus expand the cavity state described by theoperator ˆ d about the vacuum: σ = ρ | (cid:105)(cid:104) | + ( ρ | (cid:105)(cid:104) | + H.c.)+ ρ a11 | (cid:105)(cid:104) | + ( ρ | (cid:105)(cid:104) | + H.c.) + O ( (cid:15) ) . (54)The density matrix for the resonator is then given by ρ = Tr c [ σ ] = ρ + ρ + O ( (cid:15) ) . (55)where Tr c denotes the trace over the cavity mode. Fromthe master equation (Eq.(50)) we then derive the equa-tions of motion for the ρ ij . Adiabatic elmination of theoff-diagonal elements ρ and ρ (described in detailin Ref.[30]) allows us to write a closed set of equationsfor the diagonal elements ρ and ρ . The result is astochastic master equation for ρ = ρ
00 + ρ
11, which is dρ = k [ X, [ X, ρ ]] dt + (cid:112) ηk H [ X ] ρdW, (56)where the measurement strength k = 4 ˜ A /κ . Defining˜ k = 8 ηk , and making a Gaussian ansatz for the quan-tum state, we find Eqs. (25a) - (25e) for the means andvariances of the quadratures X and Y . C. Squeezing via Feedback Control
There are three formulations that can be used to ana-lyze the behavior of an observed linear quantum system:the Heisenberg picture (the input-ouput formalism), theSchr¨odinger picture (the SME) and the equivalent classi-cal formulation, introduced in Ref. [30]. We have alreadyused the first two methods in our analysis above. To an-alyze the effect of feedback we now use the third. Theequivalent classical formulation is given by the equations dx = − γ xdt + (cid:113) γ ˜ T eq dW x (57) dy = − γ ydt + (cid:113) γ ˜ T eq dW y + (cid:112) ˜ k dV + (cid:112) ˜ λ dV (58)where x and y are now classical dynamical variables, andas always the noise sources, W i and V i , are mutual un-correlated Wiener processes. We have now included twomeasurements of the x quadrature, one with strength˜ k and the other with strength ˜ λ , for reasons that willbe explained below. The measurement records (i.e. thehomodyned output signals) for these measurements aregiven by dI = (cid:112) ˜ kxdt + dU (59) dI = (cid:112) ˜ λxdt + dU . (60)0Once again the U i are mutually uncorrelated Wiener pro-cesses. Of interest are the quantities (cid:104) ¯ X (cid:105) and (cid:104) ¯ X (cid:105) ,which are (respectively) the two observers’ estimates ofthe X quadrature. Note that these are not the same as x above. When ˜ λ = 0, so that there is no second mea-surement, the equation of motion for (cid:104) ¯ X (cid:105) is naturallythat given by Eqs (25a) - (25e). With the second mea-surement, the dynamics of the means and variances forthe first observer become d ¯ X = − ( γ/
2) ¯
Xdt + (cid:112) ˜ kV X d ˜ U (61a) d ¯ Y = − ( γ/
2) ¯
Y dt + (cid:112) ˜ kCd ˜ U (61b)˙ V X = − ˜ kV X − γ ( V X − ˜ T eq ) (61c)˙ V Y = − ˜ kC + 2 k + 2 λ − γ ( V Y − ˜ T eq ) (61d)˙ C = − γC − ˜ kV X C (61e)where k = ˜ k/ (8 η ) and λ = ˜ λ/ (8 η ) are the strengths ofthe respective measurements (under the usual definitionof measurement strength [31]), and the η i are the respec-tive efficiencies of the measurements. We also introducea fourth set of noises, ˜ U i , where ˜ U appears in the aboveequations for the first observer, and ˜ U would appear inthe equations for the second observer, although we willnot need those here. The ˜ U i are given by [30] d ˜ U i = dI i − ¯ X i dt = (cid:112) ˜ k ( x − ¯ X i ) dt + dU i . (62)While it is not obvious, it turns out that the d ˜ U i arealso mutually uncorrelated, and uncorrelated with all theother noise sources.Armed with the above equations, we now introducefeedback into the system. We apply a continuous feed-back force F ( t ) = αγ ¯ X sin( ω M t ) to the system in thelab frame. Discarding rapidly oscillating terms (makinga rotating-wave approximation), this results in the fol-lowing dynamics for the system: dx = − (cid:16) γ x + αγ X (cid:17) dt + (cid:113) γ ˜ T eq dW x (63) dy = − γ ydt + (cid:113) γ ˜ T eq dW y + (cid:113) ˜ k/ dV + (cid:113) ˜ λ/ dV (64)The feedback provides a damping force on the X quadra-ture with a rate αγ/
2. Since applying a known force tothe system cannot change the observers’ uncertainty re-garding the classical coordinates, the equations of motionfor the variances for both observers remain the same. Theequations of motion for the means however also pick upexactly the same damping terms. Thus for observer onewe have ˙¯ X = − (1 + α ) γ X + (cid:112) ˜ kV X ˙˜ U (65)˙¯ Y = − γ Y + (cid:112) ˜ kC ˙˜ U . (66) We now want to calculate the variance of the X quadrature under this feedback protocol, and also thespectrum of the output signal for both observers. Sincethe X and Y quadratures are not coupled, we needmerely solve the two coupled equations˙ x = − γ x − αγ X + (cid:113) γ ˜ T eq ˙ W x (67)˙¯ X = − (cid:16) γ αγ kV X (cid:17) ¯ X + ˜ kV X x + (cid:112) ˜ kV X ˙ U , (68)where we have used Eq.(62) to write the equation for¯ X in terms of ˙ U rather than ˙˜ U . The unconditionalvariance of the X quadrature under feedback, which wewill denote by V fb X, tot , is given by the variance of x .Solving for the steady-state value of V fb X, tot using theusual techniques of Ito calculus, and using the fact that˜ T eq = V X + (˜ k/γ ) V X (c.f. Eq. (30)), we obtain Eq. (31).We see that as the feedback strength α tends to infinity, V fb X → V X , as claimed above.To calculate the spectrum of the output signal for thefirst observer we first transform Eqs (67) and (68) to thefrequency domain and solve them. The solution is of theform (cid:18) x ( ω )¯ X ( ω ) (cid:19) = M ( ω ) (cid:113) γ ˜ T eq ˙ W x ( ω ) (cid:112) ˜ kV X ˙ U ( ω ) (69)with (cid:104) ˙ W x ( ω ) ˙ W x ( ω (cid:48) ) (cid:105) = (cid:104) ˙ U ( ω ) ˙ U ( ω (cid:48) ) (cid:105) = δ ( ω + ω (cid:48) ) . (70)The output signal for the first measurement is: I ( ω ) = (cid:112) ˜ kx ( ω ) + ˙ U ( ω ) . (71)The corresponding output spectrum is defined via: (cid:104) I ( ω ) I ( ω (cid:48) ) (cid:105) = S I ( ω ) δ ( ω + ω (cid:48) ) (72)Using Eqs. (69),(70), one finds the zero-frequency spec-trum to be given by: S I (0) = (1 + α + 2˜ kV X /γ ) (1 + α ) . (73)where we have made use of Eq. (30). Referring this backto the oscillator by dividing by ˜ k results in Eq. (32).Similarly, the output of the second measurement isgiven by I ( ω ) = (cid:112) ˜ λx ( ω ) + ˙ U ( ω ) , (74)Using this definition and Eqs. (69),(70), and makinguse of Eq. (30), we find the output spectrum given inEq. (33).1 IV. CONCLUSIONS
In this paper, we have provided a thorough and fullyquantum treatment of back-action evasion using a drivenelectromagnetic cavity which is parametrically coupled toa mechanical oscillator. We have considered both the un-conditional and conditional aspects of the measurement.In particular, we have derived exactly how strong thecoupling must be to beat the standard quantum limit,and to achieve a conditionally squeezed state. We havealso shown how feedback can be used to generate truesqueezing, and how this squeezing can be detected using a second measurement.
Acknowledgments
We thank S. Girvin and K. Schwab for useful conver-sations. A.C. acknowledges the support of NSERC, theCanadian Institute for Advanced Research, and the Al-fred P. Sloan Foundation. F.M. acknowledges support byNIM, SFB 631, and the Emmy-Noether program of theDFG. [1] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.Clerk, M. P. Blencowe, and K. C. Schwab, Nature (Lon-don) , 193 (2006).[2] R. G. Knobel and A. N. Cleland, Nature (London) ,291 (2003).[3] M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab,Science , 74 (2004).[4] N. E. Flowers-Jacobs, D. R. Schmidt, and K. W. Lehnert,Phys. Rev. Lett , 096804 (2007).[5] C. H¨ohberger-Metzger and K. Karrai, Nature , 1002(2004).[6] S. Gigan, H. B¨ohm, M. Paternostro, F. Blaser, J. B.Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer,and A. Zeilinger, Nature , 67 (2006).[7] A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, andT. J. Kippenberg, Phys. Rev. Lett. , 243905 (2006).[8] J. D. Thompson, B. M. Zwickl, A. M. Jayich,F. Marquardt, S. M. Girvin, and J. G. E. Harris,arXiv:0707.1724 (2007).[9] D. Kleckner and D. Bouwmeester, Nature , 75 (2006).[10] D. H. Santamore, A. C. Doherty, and M. C. Cross, Phys.Rev. B , 144301 (2004).[11] K. Jacobs, P. Lougovski, and M. P. Blencowe, Phys. Rev.Lett. , 147201 (2007).[12] I. Martin and W. H. Zurek, Phys. Rev. Lett. , 120401(2007).[13] E. Buks, E. Arbel-Segev, S. Zaitsev, B. Abdo, and M. P.Blencowe, Europhysics Lett. , 10001 (2008).[14] K. Jacobs, A. N. Jordan, and E. K. Irish, Eprint:arXiv:0707.3803 (2007).[15] L. F. Wei, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev.Lett. , 237201 (2006).[16] A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys.Rev. Lett. , 148301 (2002).[17] A. A. Clerk and D. W. Utami, Phys. Rev. A , 042302(2007).[18] V. B. Braginsky, Y. I. Vorontsov, and K. P. Thorne, Sci- ence , 547 (1980).[19] V. B. Braginsky and F. Y. Khalili, Quantum Measure-ment (Cambridge University Press, Cambridge, 1992).[20] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sand-berg, and M. Zimmermann, Rev. Mod. Phys. , 341(1980).[21] M. F. Bocko and R. Onofrio, Rev. Mod. Phys. , 755(1996).[22] C. M. Caves, Phys. Rev. D , 1817 (1982).[23] A. A. Clerk, Phys. Rev. B , 245306 (2004).[24] C. A. Regal, J. D. Teufel, and K. W. Lehnert,arXiv:0801.1827v2 (2008).[25] A. Schliesser, R. Rivi`ere, G. Anetsberger, O. Arcizet, andT. J. Kippenberg, arXiv:0709.4036 (2007).[26] R. Ruskov, K. Schwab, and A. N. Korotkov, Phys. Rev.B , 235407 (2005).[27] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,Phys. Rev. Lett. , 093902 (2007).[28] D. F. Walls and G. J. Milburn, Quantum Optics (Springer, New York, 1995).[29] C. W. Gardiner and P. Zoller,