Noise Suppression for Micromechanical Resonator via Intrinsic Dynamic Feedback
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Noise Suppression for Micromechanical Resonator via Intrinsic Dynamic Feedback
H. Ian, ∗ Z. R. Gong, and C. P. Sun † Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing, 100080, China (Dated: October 27, 2018)We study a dynamic mechanism to passively suppress the thermal noise of a micromechanicalresonator through an intrinsic self-feedback that is genuinely non-Markovian. We use two coupledresonators, one as the target resonator and the other as an ancillary resonator, to illustrate themechanism and its noise reduction effect. The intrinsic feedback is realized through the dynamics ofcoupling between the two resonators: the motions of the target resonator and the ancillary resonatormutually influence each other in a cyclic fashion. Specifically, the states that the target resonatorhas attained earlier will affect the state it attains later due to the presence of the ancillary resonator.We show that the feedback mechanism will bring forth the effect of noise suppression in the spectrumof displacement, but not in the spectrum of momentum.
PACS numbers: 85.85.+j, 85.25.Cp, 45.80.+r
I. INTRODUCTION
Recently, interests have been generated on coolingtechniques for mechanical systems at nano- and micron-scales [1]. Among them, the typically employed is thefeedback cooling technique where an external feedbackcircuit is responsible for detecting the motion of the tar-get and feeding a counteracting force against this motion;through a general decrease of magnitude in the densitynoise spectrum, it was shown that the feedback can ef-fectively reduce the fluctuation of the target and providea cooling mechanism [3, 4]. Some experiments based onthe models that contain the feedback loops have been car-ried out in the past few years. A few are directed towardsthe cooling of micron- to nanometer-size mechanical res-onators, aiming to reach a macroscopic quantum mechan-ical ground state and serving as a powerful manifestationof quantum mechanical effects [5, 6, 7, 8]. Other experi-ments succeed in slowing down the motion of micron-sizemirrors through the radiation pressure of an optical fieldin a Fabry-P´erot (FP) cavity, aiming to reach a noiselevel and equivalently an effective temperature that arepertinent to the employment of high-precision detectionof gravity waves [9, 10, 11, 12].The forementioned implementations of feedback cool-ing through reduction of noise fluctuations invariably relyon an electrical circuitry external to the target system tobe cooled. The controller here is usually fixed and at-tracts or repels the resonator through either electrostaticCoulomb force or Lorentzian force. If such an externaldetection-control unit could be eliminated in favor of amechanism with self-detection of and self-adjustment tothe target’s thermal motion, we call the mechanism “self-cooling” [13, 15]. Devices implementing this self-coolinguse less components and are free from the reliance on an ∗ Electronic address: [email protected] † Electronic address: [email protected];URL: external circuitry and hence prone to less noise sources.In one case [13], an augmented cavity along with anextra optical cavity field is established on the other sideof the mirror, in addition to the regular FP cavity, soas to counteract the radiation pressure from the originalcavity field. This extra field cushions the motion of thepressure mirror and plays the role of feedback. In an-other case [14, 15], a set of Josephson junctions behavingas a qubit, serially connected to a mechanical resonat-ing beam, serves delayed supercurrent into the circuitaccording to the magnetic flux through the circuit loop.The magnetic flux is controlled by the vibrating motionof the beam, which in turn is controlled by the magneticfield generated by the current feed. Such a mutual de-pendence furnishes a self-feedback mechanism. It shouldbe pointed out that both of the self-feedback setups re-quire delayed feedback, which assumed a priori a non-Markovian approximation that explicitly depends on thehistory of the target’s motion. In these phenomenolog-ical treatments, the cooling target either couples itselfto a static controller and makes itself prone to the noisestemmed from the feedback, or couples to a mechanicallystatic detection construct and receives manually delayedfeedback.Hereby, we present a dynamic model based on an in-trinsic mechanism with non-Markovian feedback, whichis obviously free from an external feedback loop and doesnot rely on a presupposition of historical dependence.This mechanism is illustrated by a simple mechanical sys-tem in which the target is modeled by a harmonic oscilla-tor and attached to a dynamic controller, which is a rela-tively heavier resonator, through a spring. The target iscontrolled by an intrinsic feedback through the dynamicsof coupling: earlier positions and velocities of the targetaffects the motion of the controller and this influence issubsequently fed back to the target. Consequently, theaccumulation of earlier states of the target will affect thestate of itself later. With proper parameter setup, thetarget essentially experiences a resistance and deceleratesits motion; its displacement variance is shrunk, noise sup-pressed and effective temperature cooled down. The lack
FIG. 1: (Color online) The diagrammatic figure shows thearrangement of the three springs and the two masses. Fromleft to right, they are: the spring of constant k , the targetmass m , the spring of constant g , the ancillary mass M , andthe spring of constant K . G ( t ) is the harmonic driving force. of a specific detection device for the motion of the targetresonator and an external feedback circuit characterizesthe intrinsic nature of the mechanical feedback. Our nu-merical analysis shows the existence of a noise suppres-sion capability of our scheme, e.g. the variance of dis-placement can be reduced to 0 . × − m , and there-from a cooling capability under a practical setting acces-sible in current experiments. We note that the scheme istheoretically illustrative through its simple model setupyet widely applicable because the general oscillator sys-tems can be extended to quantum bosonic systems andother cases. In fact, a similar model and mechanism hasbeen proposed to actively cool down the torsional vibra-tion of a nanomechanical resonator through spin-orbitinteractions [16].The model will be explained in Sec.II and its delayfunction then derived a posteriori to examine its non-Markovian dependence. The complete solution of thesystem dynamics is given in Sec.III, with which we willderive the density noise spectrum and calculate the theo-retical noise suppression rate. The associated numericalresults will be presented in Sec.IV, given various param-eter setups. The analysis is extended to the domain ofmomentum noise in Sec.V. II. INTRINSIC FEEDBACK BY COUPLINGDYNAMICSA. The Model
Our model setup (see Fig.1) comprises two masses andthree springs. The two masses are denoted by m and M , respectively. The mass m is the target and typicallylighter whereas the mass M serves as an ancillary con-troller and is relatively heavier. The three springs aredenoted by their Hooke’s constants k , g and K , respec-tively. The spring of constant k attaches the lighter mass m to the fixed wall on the left and the spring of constant K attaches the heavier mass M to the fixed wall on theright. The spring of constant g strings the two masses to-gether. Such a setup, intuitively, grants the heavier mass M the function of a suspension system and a mediumfor the feedback. The symbol G ( t ) represents an exter-nal driving force which is necessary for the discussion ofcooling but can be deemed zero for the present.The connected springs will give rise to mechanical vi-brations of the masses. We let ¯ ω = p ( k + g ) /m de-note the effective mechanical resonance frequency forthe mass m , assuming the other mass M is fixed, and¯Ω = p ( K + g ) /M the equivalent for the mass M , as-suming the mass m is fixed. Besides these mechanicalvibrations, we assume each of the masses experience africtional damping and we let γ denote the damping co-efficient for the mass m and Γ that for mass M .Then according to the setup above, the coordinatesof the two masses obey a coupled system of classicalLangevin equations¨ x + γ ˙ x + ¯ ω x − gm Q = f (1)¨ Q + Γ ˙ Q + ¯Ω Q − gM x = F (2)where x is the coordinate of the mass m and Q that of themass M . f and F on the right hand side of the equationsdenote the random thermal noise generated by the mass m and M , respectively, due to their frictional damping.The frictional damping terms as dissipation and therandom noise terms as fluctuations constitute the totalBrownian thermal force in the classical Langevin formal-ism. This thermal force induces an thermal environment,the effect of which is divided among the two terms f ( t )and F ( t ) according to the fluctuation-dissipation rela-tions [17] h f ( t ) f ( t ′ ) i = 2 k B T γm δ ( t − t ′ ) , (3) h F ( t ) F ( t ′ ) i = 2 k B T Γ M δ ( t − t ′ ) . (4)For now we assume the two masses are independently in-teracting with two thermal environments. That is, thereare no correlations between the fluctuations of the twomassesActually, we can realize from Eq.(1) and Eq.(3) thatthe motion of the mass m is resisted by a frictional force γ , which in turn is transduced into thermal energy andheats up itself. This process, however, is mediated bythe mass M that stands between m and the fixed wallthrough the term ( g/M ) Q . If there were not the mass M , the kinetic oscillation of m would be instantly reactedby the surrounding springs of Hooke’s constant k and K .With the presence of the mass M and the extra springof constant g , the oscillation of the mass m will firstsqueeze the spring of constant g , and then the squeezedspring will release and push the mass M to the right. Itfollows that the spring of constant K will be squeezedsuccessively. The mediating mass M breaks the originalsingle spring into two and permits these two springs tostay in different states, squeezed and released, and henceessentially delays the reaction of the springs. This cas-caded process is then reflected by the wall and executed FIG. 2: Schematic illustration of the feedback mechanismshows the controller M receives signal x ( t ) and outputs signal Q ( t ) according to the motion of the target m . in a reversed order to the left; the delayed reaction of thesprings of constant g and K acts back onto the oscillatingmass m . The delayed reaction depends on the oscillatingvelocity of the target mass m and can thus be considereda feedback onto itself.The entire process can be regarded as a feedback loopfrom the view of control theory through the flow diagramshown in Fig.2. The target resonator m is the system tobe controlled and the ancillary resonator M becomes thecontroller which detects the signal x ( t ) as its input andfeeds the signal Q ( t ) as its output. The cycle time of theloop corresponds to the delay of the controller-to-systemreaction. The dependence of this delayed reaction onthe tunable parameters, mainly those spring constants,allows us to control this self-feedback precisely to coun-teract the oscillating motion of the target mass.The weakened motion is converted to a reduction of ef-fective temperature of the target mass through an equiv-alent relation between the autocorrelation of the tar-get’s displacement and temperature, derived from thefluctuation-dissipation relation Eq.(3). Under normalcircumstances, that is, when the coupling mass M werenot present, this equivalent relation can be written as h x ( t ) x ( t ′ ) i = 2 πk B γT ξ ( t − t ′ ) (5)where γ is defined as in Eq.(1) and ξ is a function to begiven explicitly. Experimentally, the statistical varianceof the coordinate is measured and the above relation isused to compute an effective temperature.Increasing the damping coefficient γ will certainly in-crease the enveloping rate of the oscillating motion of x ( t ). However, this reduction of motion cannot lead toan equivalent noise suppression effect upon the targetmass m because of the constraint imposed by the friction-induced fluctuation relation described above. In fact, ac-cording to this relation, increasing damping rate leads tomore thermal dissipation of the system. That is to say,simply enlarging frictional force merely results in a heat-ing effect upon the target system. Our aim, therefore,is to reduce the temperature of the target mass by re-ducing its coordinate variance through the feedback thatdoes not simultaneously increase the damping coefficient.After we derive the explicit feedback response below, wewill show the feedback is actually shrinking the variance in Sec.III. B. The Feedback
Generally, a feedback external to an oscillating systemhas the effect of adding an extra driving force term onthe right hand side of the equation of motion, i.e.¨ y + γ ˙ y + ¯ ω y = f + F fb (6)where we have let y denote the coordinate of a general dy-namic system, F fb the feedback force, and γ , ¯ ω , f termsof similar meanings to those defined in Eq.(1). If the tar-get were to be cooled, the motion of the system shouldbe slowed down. In other words, behaving as a func-tion of the target’s velocity, the force F fb should have anequivalent effect of increasing the damping coefficient,but do not increase the fluctuations. Besides, the feed-back force should be dependent on the target’s positionand the noise source.Summing up these requirements, we expect F fb to bea function of y ( t ), ˙ y ( t ) and f . To overcome the genericfluctuation-dissipation relations and hence accomplish anefficient noise suppression, we shall use a non-Markoviantype feedback: F fb not only depends on the current valueof the velocity and the position of the target, but alsotheir historical values at past times. If we represent thehistorical dependence by a time-derivative d K ( y, ˙ y, τ ) / d τ and weigh the contribution of the histories by a delayfunction h ( t − τ ), where t stands for the current timeand τ for the time in the past, the feedback force can bewritten as an integral with respect to τ , F fb = Z t −∞ d τ d K ( y, ˙ y, τ )d τ h ( t − τ ) . (7)The above formula shows the mathematical characterof a general feedback force. Inversely, any function thatcan express the same character should be considered afeedback force. Therefore, we can verify the dynamicresponse of the coupling mass M in our model as an ef-fective feedback mechanism by finding the correspondingspecific expression for K ( y, ˙ y, t ) and h ( t − τ ). To do so, wesolve Eq.(2) by using Fourier transforms and integrationby parts (the detailed derivation is given in Appendix A) Q ( t ) = 12 π Z ∞−∞ d τ d ω gM x ( τ ) + F ( τ )( ¯Ω − ω ) + i Γ ω e − iω ( τ − t ) (8)= 1¯Ω φ ( x, t ) + 2 √ − Γ × (9) Z t −∞ d τ d φ ( x, τ )d τ h ( t − τ )where φ ( x, t ) = gM x ( t ) + F ( t ) (10)denotes the inhomogeneous part of the equation, i.e. theexternal driving force to M . In the solution, h ( ξ ) = −
12 ¯Ω exp (cid:20) −
12 Γ( ξ ) (cid:21) np − Γ × (11)cos (cid:20) ξ p − Γ (cid:21) + Γ sin (cid:20) ξ p − Γ (cid:21)(cid:27) is the delay function that we look for and K ( x, τ ) = 2 √ − Γ φ ( x, τ ) (12)correspondingly shows the historical dependence.If we plug the solution (9) into Eq.(1), i.e. reduce thedegree of freedom of the variable Q ( t ) in the equationby substituting with its formal solution, we arrive at anintegro-differential equation of only a single variable x ( t ),which is comparable to the feedback-containing equationof motion that appears in previous literature [12, 15].However, in the latter case the delay function is assumeto be a non-Markovian approximation h ( ξ ) = 1 − e − rξ (13)and K ( y, ˙ y, τ ) is left unknown. Here, we have explicitlyimplemented a self-feedback mechanism that exerts forceback onto the mass m after certain delay through theuse of the coupling mass M . The sinusoidal factor inthe delay Eq.(11) illustrates the damped oscillating mo-tion of the mass M . The implicit time derivative of x ( t )within the integrand implies up to an equivalent effect anadditional friction to the motion of the mass m , whichdamps the oscillation without increasing the target’s fluc-tuation. The non-integral term is Markovian and has thesame order as ¯ ω x . Though it does not appear in Eq.(7),this term effectively reduces the oscillating frequency andshall not counteract the feedback effect. Therefore, bothterms impose noise suppression effect to our target. Theonly limitation, however, is the thermal fluctuation F ( t )from the coupling mass M itself and it will result in alimit for the suppression because of the competition be-tween this fluctuation and the effective feedback. III. EXACT SOLUTIONS OF THE LANGEVINEQUATIONS
In order to examine the validity of the above proposedmechanism and verify the efficacy of the cooling rate, wefind the analytic solutions of the Langevin equations (1)and (2) to reflect the displacement of the target m as aresponse to its own thermal noise, the motion of the an-cillary mass M and the thermal noise of mass M . Fromthe response function, we shall derive the effective damp-ing coefficient and vibrating frequency of the target mass m as well as the autocorrelation function of its coordi-nate. The noise spectrum and the total noise fluctuationare then defined upon these derived quantities. A. The Noise Spectrum
The displacement spectrum can be written as the sumof responses of the noise terms (the derivation is given inAppendix B)˜ x ( ω ) = L f ( ω ) ˜ f ( ω ) + L F ( ω ) ˜ F ( ω ) . (14)Note that the two different susceptibilities L f ( ω ) = 1¯ ω e − ω + iωγ e (15)and L F ( ω ) = g/m (cid:2) ¯Ω − ω + iω Γ (cid:3) [¯ ω e − ω + iωγ e ] (16)reflect the system’s different responses to thermal fluctu-ations from the mass m and the mass M . Here, we havedefined the effective vibrating frequency of the mass m to be ¯ ω e ( ω ) = ¯ ω − g mM ¯Ω − ω ( ¯Ω − ω ) + ω Γ (17)and the effective damping coefficient of the mass m to be γ e ( ω ) = γ + g mM Γ( ¯Ω − ω ) + ω Γ . (18)Eq.(14) means that the mechanical susceptibility or re-sponse function of the target m is adjusted because of thedynamic coupling of the target mass m to the ancillarymass M . The first term represents the direct effect of thethermal bath acting on the target mass m ; whereas thesecond term represents the indirect effect of the thermalbath onto the mass m through the mediating mass M and the coupling between the two masses.The positivity of the second term in Eq.(18) has as-serted our expectation of increasing the damping ratewithout increasing thermal force. We shall also notethat an additional noise source F ( ω ) is imposed ontothe adjusted susceptibility Eq.(15) due to the dynamiccoupling. But seeing that it is divided by a frequency-squared term, we expect it to be negligible when the tar-get mass m is not resonating at a frequency close to thatof the mass M .The density noise spectrum (DNS), which is definedby the equation S x ( ω ) = 12 π Z ∞−∞ d ω ′ h ˜ x ( ω )˜ x ( ω ′ ) i , (19)can thus be computed from Eq.(14) S x ( ω ) = 2 k B Tm γ e (¯ ω e − ω ) + ω γ e . (20)Comparing the above equation with the case when M isabsent, i.e. when γ e is degenerated to γ , we observe ageneral suppression at the noise peak and spreading ofthe noise spectrum. B. Noise Suppression
The observable effect of the noise sources on the mo-tion of the mass m is equivalent to the variance of thedisplacement of the mass m in time domain, which is de-fined as the average of the entire noise spectrum, i.e. theintegral of the DNS of x , (cid:10) x ( t ) (cid:11) = 12 π Z ∞−∞ d ω S x ( ω ) (21)= k B Tmπ Z ∞−∞ γ e (¯ ω e − ω ) + ω γ e d ω. (22)The integral is computable after we approximate the ef-fective damping rate and vibrating frequency by truncat-ing their expansions shown in Appendix C. As a result,the variance is a variable of temperature and the springconstants (cid:10) x (cid:11)(cid:12)(cid:12) k,K,g = k B T g + Kg ( k + K ) + kK . (23)The noise fluctuation depends on the three spring con-stants k , K and g as its parameters, and is independentof the masses m and M and the damping rate γ and Γ ofthe resonators. We shall notice that the noise suppres-sion effect for the target resonator is always present forall values the spring constant g takes. This ideal result isdue to the non-Markovian feedback we derived in Sec.II,which always increases the effective damping while re-taining the same fluctuations.Eq.(23) shows a complex relation between itself andits three parameters of spring constants. To illustrate itsbehavior, we focus on its relation with the spring constant g . It is a monotonic decreasing function of g and its twolimiting values are (cid:10) x (cid:11)(cid:12)(cid:12) g → = k B Tk (24) (cid:10) x (cid:11)(cid:12)(cid:12) g →∞ = k B Tk + K , (25)which coincides with our expectation that enlarging theconstant g will render the feedback more effective due tothe enlarged feedback amplitude Eq.(10).The symmetry between the target mass and the ancil-lary mass in the model setup allows us to compute thevariance of the ancillary mass following the same method-ology (cid:10) Q (cid:11)(cid:12)(cid:12) k,K,g = k B T g + kg ( k + K ) + kK . (26)The limiting value of (cid:10) Q (cid:11) at g → ∞ is identical to thatof (cid:10) x (cid:11) , which shows that under the extremal case wherethe two oscillating masses combines into one by a rigidbody the limiting behaviors of the two bodies become thesame. IV. NUMERICAL ANALYSIS OF NOISESUPPRESSION
To show completely the noise suppression behavior ofthe two oscillating masses, the numerical analysis is sep-arated into two parts with each part for each extremalend of the values of the spring constants whereas theother parameters are set to laboratory accessible valuesfor common micromechanical resonators. The first case,the identical case, is where the springs attaching the twomasses to the walls share the same Hooke’s constant, i.e. k = K . We examine how the noise suppression behav-ior is affected by varying the value of the constant g ofthe middle spring. The second case, the large detuningcase, occurs when the springs attaching the two massesto the walls take vastly different values of their Hooke’sconstant. Again we examine the noise suppression limitfor different values taken for the constant g of the middlespring. A. The Identical Case
We assume the springs with one end fixed to walls have k = K = 1 N m − . The lighter target mass has m =1 × − kg and its frictional damping rate γ = 0 . s − .The heavier ancillary mass has M = 1 × − kg andits frictional damping rate Γ = 4 s − . When free fromthe stringing spring g , the target mass will oscillate ata natural frequency of 10 kHz and the ancillary mass at1 kHz . The system’s initial temperature is set to roomtemperature T = 295 K .We first look at the density noise spectrum of the ancil-lary resonator when the middle spring is set to have itsconstant g = 0 . N m − , 0 . N m − , 1 N m − , 10 N m − and 100 N m − as shown in Fig.3. The noise peaks at thefrequencies ω c = 1005 Hz , 1044 Hz , 1223 Hz , 1376 Hz and1404 Hz .We notice that when tuning the spring constant g ,not only the peaking frequency is shifted to the right,the peak amplitude is reduced along with the increasedvalue of g . This proves a general suppression in noiseand an equivalent cooling effect to the system. The de-tails of the spectrum and the spread width can be shownmore apparently when we cluster the peaks together witha common frequency ω c for their corresponding peakingfrequencies, which is shown in Fig.4. The total noise fluc-tuation reached after suppression can be computed fromthe area under each curve in the figure, using Eq.(22).The attenuated noise levels corresponding to the 5 val-ues of the spring g are, respectively, 4 .
03, 3 .
73, 2 .
72, 2 . .
04 times a common factor of 10 − m .The cooling effect for the target mass m is more obvi-ous if we examine the clustered peak plot of the targetmass’ density noise spectrum shown in Fig.5. However,differing from the behavior of the ancillary mass shown inFig.3, the target mass is resonant at two peak frequenciesfor each of the 5 values of the spring constant g . Among g(N/m) (10 -21 m ) 0.01 4.03 0.1 3.73 1 2.72 10 2.12 100 2.04 T=295K S Q / k B /Hz FIG. 3: (Color online) Plot of DNS of the ancillary resonator M with parameters: k = K = 1 Nm − , M = 1 × − kg , m =1 × − kg , Γ = 4 s − , γ = 0 . s − , T = 295 K . Curves fromtop to bottom are plotted from g = 0 . Nm − , 0 . Nm − ,1 Nm − , 10 Nm − , 100 Nm − . g(N/m)
(10 -21 m ) 0.01 4.03 0.1 3.73 1 2.72 10 2.12 100 2.04 C +4 C -4 C S Q / k B /HzT=295K FIG. 4: (Color online) Rescaled plot of DNS of the an-cillary resonator M with parameters: k = K = 1 Nm − , M = 1 × − kg , m = 1 × − kg , Γ = 4 s − , γ = 0 . s − , T = 295 K and peaking frequencies shifted to the centerof the plot. Curves from top to bottom are plotted from g = 0 . Nm − , 0 . Nm − , 1 Nm − , 10 Nm − , 100 Nm − . the pairs of peaking frequencies, one group clusters inthe low frequencies and the other spreads out in the highfrequencies.The low frequency group, the rescaled along peak cen-ter plot shown on the left of Fig.5, shares exactly thesame peaking frequencies as those of the ancillary massand we expect this behavior takes place when the targetmass is resonating with the ancillary mass. This har-monic driven noise associates with the noise source ˜ F ( ω )in Eq.(14) and, as we argue before, does not contributemuch to the overall thermal noise. The high frequencygroup spreads out to peak frequencies ω c = 10 . kHz ,10 . kHz , 14 . kHz , 33 . kHz and 101 kHz for the varyingspring constant g . The anharmonic noise with respect tothe ancillary mass associates with the noise source ˜ f ( ω )in Eq.(14). Our cooling mechanism in this identical case g(N/m)
03, 3 .
73, 2 .
72, 2 .
12 and 2 .
04 times a common factorof 10 − m , respectively, identical to the values of theancillary mass. B. The Large Detuning Case
We assume, in this case, the spring constants K =1000 N m − and k = 1 N m − ; the masses M = 1 × − kg and m = 1 × − kg ; thus the natural oscillating fre-quencies for the two masses are retained. The damp-ing coefficients and the initial temperature are left un-changed. Varying the spring constant of the middlespring g over the same five values gives the noise spec-trum plot of the target mass m shown in Fig.6. Theplot is again rescaled to the center along the peakingfrequencies ω c = 10 . kHz , 10 . kHz , 14 . kHz , 33 . kHz and 100 . kHz .We note that the noise peaks at one frequency for eachvalue of the spring g . These peaking frequencies are closeto those in the identical case above but the noise sup-pression rate, as we have expected, is much better. Thatmeans the large detuning not only helps suppress thenoise source stemmed from the coupling mass M to neg-ligible amplitude but also makes the feedback more ef-fective for countering the target’s noise. The 5 values ofthe spring constant g corresponds to noise fluctuations of4 .
03, 3 .
70, 2 .
03, 0 .
37 and 0 .
04 times a common factor of10 − m .Fig.7 shows the plot of the ancillary resonator’s noisespectrum, again rescaled to the center along the peakingfrequencies. These peaking frequencies are very close tothose of the target mass. We predict that the large de-tuning between the springs K and k puts the ancillarymass M into a particularly passive role that reflects the
048 048 g(N/m)
998 999 1000 1001 10020.050.060.070.080.090.100.110.120.130.140.15998 999 1000 1001 10020.050.060.070.080.090.100.110.120.130.140.15 g(N/m) (10 -21 m ) 0.01 4.07 0.1 4.07 1 4.07 10 4.07 100 4.07 T=295K S Q / k B /Hz FIG. 7: (Color online) Rescaled plot of DNS of the ancillaryresonator M with parameters: K = 1000 Nm − , k = 1 Nm − , M = 1 × − kg , m = 1 × − kg , Γ = 4 s − , γ = 0 . s − , T =295 K and peaking frequencies shifted to the center of the plot.Curves from left to right are plotted from g = 0 . Nm − ,0 . Nm − , 1 Nm − , 10 Nm − , 100 Nm − . minute motion of the target. This also helps explain whythe noise suppression is especially effective in this case bythe fact that the mass M has its speed comparable to thetarget m but in an opposite direction such that the forceit exerts through the spring g can favorably counteractthe motion of the target m . Nonetheless, the reinforcedrole that the ancillary mass M plays means that itselfdoes not belong to the target system. As shown in thefigure, the ancillary mass almost retains its original noiselevel throughout the varying values of the spring g . V. NOISE IN MOMENTUM SPACEA. The Model with Harmonic Driving Force
The effective temperature is usually defined accordingto the equipartition theorem through the equality be-tween the thermal energy and the mechanical potentialenergy. The intrinsic nature of our model renders thestored potential energy of the target and of the environ-ment in their shared link, the middle spring g , inseparableand thus forbids our discussion of the effective tempera-ture in the coordinate space. To make our model relevantto the discussion of cooling, we examine the system dy-namics in its momentum space and define the effectivetemperature according to the kinetic energy, i.e. the mo-mentum fluctuation, of the target.In addition, to follow the convention along previousliterature, a harmonic driving force term G ( t ) = F e − iω t , (27)where ω is the driving frequency, should be added tothe feedback loop (Cf. Fig.1). In our case, this harmonicforce appears as an extra term in the equation of motionof the ancillary resonator M .Following the above arguments, we rewrite the coupledsystem of classical Langevin equations Eq.(1) and Eq.(2)as ˙ p + γp + m ¯ ω x − gQ = mf (28)˙ P + Γ P + M ¯Ω Q − gx = M ( F + G ( t )) (29)where p = m ˙ x and P = M ˙ Q denote, respectively, themomenta of the target resonator and the ancillary res-onator. The thermal fluctuations f and F still obey thesame set of relations Eq.(3) and Eq.(4).Consequently, we have a modified linear response (Cf.Eq.(14), Eq.(15) and Eq.(16)) after combining the systemof equations in the frequency domain (see Appendix D)˜ p ( ω ) = iωm h L f ( ω ) ˜ f ( ω ) + L F ( ω )[ ˜ F ( ω )+ 2 πF δ ( ω + ω )] i (30)where L f ( ω ) and L F ( ω ) are the susceptibilities Eq.(15)and Eq.(16). We can thus arrive at the noise spectrumin momentum, after following the same routine of com-putations, S p ( ω ) = 12 π Z ∞−∞ d ω ′ h ˜ p ( ω )˜ p ( ω ′ ) i (31)= 2 k B T m ω γ e (¯ ω e − ω ) + ω γ e +2 πF ω m δ ( ω + ω ) L F ( ω ) L F ( − ω ) (32) B. The Effective Temperature
The square of momentum of the target is defined sim-ilarly as in Eq.(22) (cid:10) p (cid:11) = 12 π Z ∞−∞ d ω S p ( ω ) . (33)which can be separated into two parts: the variance ofmomentum, i.e. the fluctuation or noise in momentumspace, D (∆ p ) E = k B Tπ m Z ∞−∞ d ω ω γ e (¯ ω e − ω ) + ω γ e (34)and the mean squared, i.e., the square of the steady statevalue of momentum, h p i = F ω m L F ( ω ) L F ( − ω ) . (35)Note that the harmonic force term only contributes tothe first moment of the momentum.Since thermal dissipation only induces variance of thetarget’s momentum, the effective temperature of the tar-get can inversely be determine by the variance of thekinetic energy through the energy equipartition theorem T eff = D (∆ p ) E k B m . (36)Eq.(34) is independent of the two parameters F and ω of the harmonic driving force. We conclude that the har-monic force bears no effect in reducing the system’s effec-tive temperature and hence in cooling. Eq.(34) is also notanalytically integrable, numerical integration shows theintegral takes value similar to the case when the springconstant g = 0. That means the feedback is only effectivein the displacement domain but not in the momentumdomain. VI. DISCUSSION AND REMARKS
In summary, we have proposed a theoretical model todemonstrate the general self-feedback type noise suppres-sion technique through the coupling between the targetresonator and an adjuvant system. In particular, sucha self-feedback is achieved through the dynamics of theadjuvant system. The explicit delay function has beengiven and its efficacy in reducing the noise spectrum ver-ified. We have also used numeral results to confirm ourobservations.Before concluding this paper, we add some remarks asfollows:First, the theory of thermal equilibrium state can beused to explain why the system comprising two resonat-ing masses has its noise reduced to a limit when thespring constant g tends from zero to infinity. If g werezero, the stringing spring between the two masses would disappear and the system would comprise two indepen-dent resonators, each of which will attain its thermalequilibrium over time. In this case, the system has twodegrees of freedom. According to the energy equipar-tition theorem, the system must contain twice as muchenergy as k B T /
2. For the other limit, when g tends toinfinity, the two resonators become a rigid body and pos-sess only one degree of freedom, which means the systemwould contain only k B T / g .However, how each resonator in the combined systemreduces its own energy is an open question though wecan intuitively think the heat bath may play crucial rolein the asymmetric thermalization of the two resonatorsas tow open system. Another remark is that the energyequipartition theorem is applicable only after the systementers thermal equilibrium state. When interaction oc-curs between the two degrees of freedom contributed bythe two masses, such an illustration is not appropriate.This is the reason why the nonlinear character of thenoise suppression rate cannot be explained by the energyequipartition theorem alone.In addition, we have assumed there does not exist cor-relation between the two noise sources for the two masses,i.e. h f ( t ) F ( t ′ ) i = 0. However, when the two masses oscil-late very closely with each other, the above assumptionbased on independent thermal environments will not holdand we need to consider the case where h f ( t ) F ( t ′ ) i 6 = 0 . (37)Another setting we shall consider is when the motionsof the two masses are quantized: we need to change thefluctuation-dissipation relations Eq.(3) and Eq.(4) to h f ( t ) f ( t ′ ) i ∝ W ( t, t ′ , ~ ) , (38) h F ( t ) F ( t ′ ) i ∝ U ( t, t ′ , ~ ) . (39)where W and U are not simply δ -functions and dependson the Planck constant ~ . We will present the generalinvestigation on the case with quantum fluctuations andthermal bath correlations, but here we have concentratedon the simple case. Acknowledgments
The authors thank Yong Li of University of Baseland Ying Dan Wang of NTT Basic Research Labora-tories for helpful discussions. This work is supportedby the NSFC with Grants No.90203018, No.10474104,and No.60433050. It is also funded by the NationalFundamental Research Program of China with GrantsNo.2001CB309310 and No.2005CB724508.
APPENDIX A: DERIVATION OF THEFEEDBACK DELAY FUNCTION
Fourier transforming Eq.(2) gives˜ Q ( ω ) = g ˜ x ( ω ) /M + ˜ F ( ω )¯Ω − ω + iω Γ (A1)where we use ˜ x , ˜ Q and ˜ F to denote the Fourier transformsof x , Q and F , respectively,˜ x ( ω ) = Z ∞−∞ d t x ( t ) e − iωt , (A2)˜ Q ( ω ) = Z ∞−∞ d t Q ( t ) e − iωt , (A3)˜ F ( ω ) = Z ∞−∞ d t F ( t ) e − iωt . (A4)Substituting the first two transform of the above intoEq.(A1) and taking the inverse Fourier transform of ˜ Q ,we get Q ( t ) = 12 π Z ∞−∞ d τ d ω gx ( τ ) /M + F ( τ )( ¯Ω − ω ) + i Γ ω e − iω ( τ − t ) . (A5)The linear integral with respect to ω can be viewed as acontour integral along a semicircle in the upper half-planewith two poles at ω ± = [ i Γ ± √ − Γ ] /
2. AssumingΓ ≪ ¯Ω, the corresponding residues are, R ± ( t ) = ∓ √ − Γ θ ( t − τ ) × (A6)exp (cid:20) − (cid:16) Γ ∓ i p − Γ (cid:17) ( t − τ ) (cid:21) where θ ( t − τ ) denotes the unit step function originatedfrom the positive locus of the integration path. UsingCauchy’s theorem, Q ( t ) is reduced to a single-integralform Q ( t ) = 2 √ − Γ Z t −∞ d τ φ ( x, τ ) × (A7)exp (cid:20) −
12 Γ( t − τ ) (cid:21) sin (cid:20) p − Γ ( t − τ ) (cid:21) where we have used the shorthand φ ( x, t ) as defined inEq.(10). To write Q ( t ) in our desired form, we further integrate by parts with respect to ( t − τ ) Q ( t ) = − √ − Γ Z ∞ d( t − τ ) φ ( x, t ) (A8) × exp (cid:20) −
12 Γ( t − τ ) (cid:21) sin (cid:20) t − τ p − Γ (cid:21) = 1¯Ω φ ( x, t ) + 2 √ − Γ × (A9) Z ∞ d( t − τ ) d φ ( x, τ )d( t − τ ) 2 exp (cid:20) −
12 Γ( t − τ ) (cid:21) × exp h i √ − Γ ( t − τ ) i Γ − i √ − Γ +exp h − i √ − Γ ( t − τ ) i Γ + i √ − Γ where the factor in the last two lines constitute the func-tion h ( t − τ ) as defined in Eq.(11). APPENDIX B: DERIVATION OF THERESPONSE FUNCTION AND THE NOISESPECTRUM
To find the response function, we first take the Fouriertransform of Eq.(2) and Eq.(1) to get, respectively,Eq.(A1) and ˜ x ( ω ) = g ˜ Q ( ω ) /m + ˜ f ( ω )¯ ω − ω + iωγ (B1)where we let ˜ x , ˜ Q and ˜ f denote the Fourier transformsof x , Q and f as we did in Appendix A. SubstitutingEq.(A1) into Eq.(B1), we have˜ x ( ω ) = gm g ˜ x ( ω ) /M + ˜ F ( ω )¯Ω − ω + iω Γ + ˜ f ( ω ) ! (¯ ω − ω + iωγ ) − (B2)Reshuffling the terms and putting again ˜ x ( ω ) on one handside, the equation becomes˜ x ( ω ) = ˜ f ( ω ) + g ˜ F ( ω ) / [ m ( ¯Ω − ω + iω Γ)]¯ ω − ω + iωγ − g / [ mM ( ¯Ω − ω + iω Γ)] . (B3)To find explicitly the response function, we group to-gether the real terms and the imaginary terms in thedenominator to write ˜ x ( ω ) in a familiar form similar tothe case where the coupling mass M were not present;whence the denominator becomes (cid:20) ¯ ω − g mM ¯Ω − ω ( ¯Ω − ω ) + ω Γ (cid:21) − ω + iω (cid:20) γ + g mM Γ( ¯Ω − ω ) + ω Γ (cid:21) (B4)0and we can define the terms in the two brackets as inEq.(17) and Eq.(18).Using Eq.(14), the associated fluctuation-dissipationrelations of Eq.(3) and Eq.(4) in the frequency domain D ˜ f ( ω ) ˜ f ( ω ′ ) E = 4 πk B T γm δ ( ω + ω ′ ) , (B5) D ˜ F ( ω ) ˜ F ( ω ′ ) E = 4 πk B T Γ M δ ( ω + ω ′ ) , (B6)and the independence between the two noise sources, wefind by using definition Eq.(19) S x ( ω ) = 2 k B T (cid:20) γm L f ( ω ) L f ( − ω ) + Γ M L F ( ω ) L F ( − ω ) (cid:21) (B7)where L f ( ω ) and L F ( ω ) are defined in Eq.(15) andEq.(16). Seeing that L F ( ω ) = g/m ¯Ω − ω + iω Γ L f ( ω ) , (B8)we can factor out L f ( ω ) L f ( − ω ) /m from the bracket inEq.(B7) S x ( ω ) = 2 k B Tm L f ( ω ) L f ( − ω ) × (B9) (cid:20) γ + g Γ /mM ¯Ω − ω + iω Γ 1¯Ω − ω − iω Γ (cid:21) = 2 k B Tm (cid:20) ω e − ω ) + ω γ e (cid:21) × (B10) (cid:20) γ + g mM Γ( ¯Ω − ω ) + ω Γ (cid:21) which equals to Eq.(20). APPENDIX C: DERIVATION OF THEVARIANCE OF DISPLACEMENT
The integrand of Eq.(22) has highest order of ω in thedenominator and can not be integrated analytically. Tomake the integrand integrable, we introduce a parameter µ ( K ) = g/ ( g + K ) as a variable of the spring constant K and approximate the integrand by expanding ¯ ω e withrespect to µ up to 2nd order in the vicinity of µ = 0, inother words, when K → ∞ . That is, we let¯ ω e ( µ ) ≈ ¯ ω e (cid:12)(cid:12) µ =0 + d¯ ω e d µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 µ + 12 d ¯ ω e d µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 µ . (C1)To calculate the three terms in the expansion, we firstsubstitute K with g (1 − µ ) /µ in Eq.(17),¯ ω e ( µ ) = ¯ ω − g mM gM − µ − − ω ( gM − µ − − ω ) + ω Γ . (C2)Since the lowest-order term in the numerator and thedenominator is, respectively, µ − and µ − , the second term of the above formula goes to 0 when µ →
0. Thuswe find the first term in the expansion ¯ ω e (cid:12)(cid:12) µ =0 = ¯ ω . Thefirst-order derivative of ¯ ω e ( µ ) readsd¯ ω e d µ = g mM µ − (cid:2) − ( gM − µ − − ω ) + ω Γ (cid:3) [( gM − µ − − ω ) + ω Γ ] . (C3)Again by comparing the coefficients of the lowest-orderterms in the numerator and the denominator, we shall seethe second term in the expansion d¯ ω e / d µ | µ =0 = − g/m .The third term in the expansion can be obtained similarlyby observing the limiting behavior of the numerator andthe denominator of the second-order derivative of ¯ ω e ( µ )d ¯ ω e d µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 = lim µ → g mM − gM − µ − ( gM − µ − − ω ) [( gM − µ − − ω ) + ω Γ ] , (C4)which reads − M/m ) ω after taking the limit. Com-bining the results, we have¯ ω e ≈ ¯ ω − gm µ − Mm ω µ . (C5)The truncation error rate introduced in this approxi-mation, by comparing Eq.( C2) and Eq.(C5), iserf = ∆¯ ω e ¯ ω e = ¯ ω e − (cid:2) ¯ ω − ( gµ − M ω µ ) /m (cid:3) ¯ ω e . (C6)Expanding ¯ ω e and µ giveserf = (cid:26)(cid:20) g g + K + M ω g ( g + K ) (cid:21) ( F + M ω Γ ) − g F (cid:27) × (cid:2) ( g + k )( F + M ω Γ ) − g F (cid:3) − (C7)where we have used the shorthand F = g + K − M ω .Since we only consider the usual cases with Γ ≪ ω , theterms containing the damping coefficient can be omittederf = (cid:2) g / ( g + K ) + M ω g / ( g + K ) (cid:3) F − g ( g + k ) F − g . (C8)Expanding F and multiplying the numerator and denom-inator by ( g + K ) , we find (cid:12)(cid:12)(cid:12)(cid:12) ∆¯ ω e ¯ ω e (cid:12)(cid:12)(cid:12)(cid:12) = g M ω ( g + K ) [ g ( k + K − M ω ) + k ( K − M ω )](C9)and at the other extreme of the expansion µ = 1 whence K = 0 and g can take any valueerf | K =0 = M ω g ( k − M ω ) − kM ω . (C10)Therefore, the error can be sufficiently suppressed if welet g → ∞ and the expansion of ¯ ω e around µ = 1 isvalidated.Following the same reasoning and procedure, we canapproximate the effective damping coefficient Eq.(18), γ e ≈ γ + M Γ m µ (C11)1which becomes ω -independent. Substituting Eq.(C5) andEq.(C11) into Eq.(22), we get, after minor algebra, (cid:10) x ( t ) (cid:11) = k B T γ ′ mπγ e Z ∞−∞ ω ′ − ω ) + ω γ ′ d ω (C12)where γ ′ = γ + ( M Γ /m ) µ ( M/m ) µ + 1 (C13)¯ ω ′ = s ¯ ω − ( g/m ) µ ( M/m ) µ + 1 (C14)can be considered the effective damping coefficient andvibrating frequency of the mass m after the approxima-tion. We can apply a limiting process to Eq.(C12) (cid:10) x ( t ) (cid:11) = k B T γ ′ mπγ e lim t → Z ∞−∞ e − iωt (¯ ω ′ − ω ) + ω γ ′ d ω (C15)and compute the integral using the theorem of residuesas we did in Appendix A (cid:10) x ( t ) (cid:11) = k B T γ ′ mγ e ¯ ω ′ (C16)= k B T g + Kg ( k + K ) + kK . (C17) APPENDIX D: DERIVATION OF EFFECTIVETEMPERATURE