Normal--mode splitting in a weakly coupled optomechanical system
Massimiliano Rossi, Nenad Kralj, Stefano Zippilli, Riccardo Natali, Antonio Borrielli, Gregory Pandraud, Enrico Serra, Giovanni Di Giuseppe, David Vitali
NNormal–mode splitting in a weakly coupled optomechanical system
Massimiliano Rossi,
1, 2, ∗ Nenad Kralj, Stefano Zippilli,
2, 3
Riccardo Natali,
2, 3
Antonio Borrielli, Gregory Pandraud, Enrico Serra,
5, 6
Giovanni Di Giuseppe,
2, 3, † and David Vitali
2, 3, 7, ‡ School of Higher Studies “C. Urbani”, University of Camerino, 62032 Camerino (MC), Italy School of Science and Technology, Physics Division, University of Camerino, 62032 Camerino (MC), Italy INFN, Sezione di Perugia, 06123 Perugia (PG), Italy Institute of Materials for Electronics and Magnetism,Nanoscience-Trento-FBK Division, 38123 Povo (TN), Italy Delft University of Technology, Else Kooi Laboratory, 2628 Delft, The Netherlands Istituto Nazionale di Fisica Nucleare, TIFPA, 38123 Povo (TN), Italy CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy (Dated: January 8, 2018)Normal–mode splitting is the most evident signature of strong coupling between two interacting subsystems.It occurs when two subsystems exchange energy between themselves faster than they dissipate it to the envi-ronment. Here we experimentally show that a weakly coupled optomechanical system at room temperaturecan manifest normal–mode splitting when the pump field fluctuations are anti-squashed by a phase-sensitivefeedback loop operating close to its instability threshold. Under these conditions the optical cavity exhibits ane ff ectively reduced decay rate, so that the system is e ff ectively promoted to the strong coupling regime. Keywords: cavity optomechanics, active feedback, squashed states, strong coupling regime, normal mode spltting
Normal–mode splitting is the hallmark of strongly coupledsystems. In this regime two interacting systems exchangeexcitations faster than they are dissipated, and form collec-tive normal modes the hybridized excitations of which aresuperpositions of the constituent systems’ excitations [1, 2].This regime is necessary for the observation of coherent quan-tum dynamics of the interacting systems and is a centralachievement in research aimed at the control and manipulationof quantum systems [3]. In cavity opto / electro–mechanics,where electromagnetic fields and mechanical resonators inter-act via radiation pressure, normal–mode splitting and strongcoupling have already been obtained, using su ffi ciently strongpower of the input driving electromagnetic field [4], or work-ing at cryogenic temperatures with relatively large single-photon coupling [5, 6].In this letter we report on the oxymoron of observingnormal–mode splitting in a weakly coupled system. Specif-ically, we have designed and implemented a feedback sys-tem [7, 8] which permits the formation of hybridized normalmodes also at room temperature and in a relatively modestdevice, in terms of single-photon optomechanical interactionstrength (as compared to the devices used in Refs. [4–6]). Oursystem is basically weakly coupled at the driving power thatwe can use (limited by the onset of optomechanical bistabilityat stronger power), and the emergence of hybridized optome-chanical modes is observed when the light amplitude at thecavity output is detected and used to modulate the amplitudeof the input field driving the cavity itself. The feedback worksin the anti-squashing regime, close to the feedback instability,where light fluctuations are enhanced over a narrow frequencyrange around the cavity resonance. In this regime the systembehaves e ff ectively as an equivalent optomechanical systemwith reduced cavity linewidth. This allows coherent energyoscillations between light and vibrational degrees of freedomwhen, for example, a coherent light pulse is injected into the cavity mode, similar to what has been discussed in Ref. [6].Light (anti–) squashing [9–11] refers to an in–loop (en-hancement) reduction of light fluctuations within a (positive)negative feedback loop. Even if the sub-shot noise features ofin-loop light disappear out of the loop, so that squashing is dif-ferent from real squeezing [9], useful applications of in-looplight have been proposed [10, 11] and realized [7, 8]. In thiscontext, the results presented here demonstrate the potentialityof the in-loop cavity as a novel powerful tool for manipulatingmechanical systems. It can be useful in situations which re-quire a reduced cavity decay or when, due to technical limita-tions, increasing the pump power is not a viable option, e.g. incase of optomechanical bistability (as in our system) or largeabsorption (which may lead to detrimental thermorefractivee ff ects, in turn detuning the cavity mode [12]). Our resultsapply directly to the high-temperature classical regime. How-ever, as already discussed in the case of ground state cool-ing [7], this technique can also be successfully applied to thecontrol of mechanical resonators at the quantum level.Our system, described in more detail in Refs. [7, 8, 13],consists of a double–sided, symmetric, optical Fabry–P´erotcavity and a low–absorption [13] circular SiN membrane in amembrane–in–the–middle setup [14]. We focus on the fun-damental mechanical mode, with resonance frequency ω m = π × .
13 kHz and a decay rate γ m = π × .
18 Hz [7, 8].The cavity has an empty–cavity finesse of F = κ = π ×
20 kHz [7, 8].Experimentally, these values are determined by placing themembrane at a node (or an anti–node) of the cavity standingwave, since the finesse is generally diminished by the mem-brane optical absorption and surface roughness, and is a peri-odic function of its position [13, 15].The experimental setup is shown in Fig. 1. Two laser beamsare utilized. The probe beam is used both to lock the laser fre-quency to the cavity resonance and to monitor the cavity phase a r X i v : . [ qu a n t - ph ] J a n HomodynedetectorLaser Probe beamCooling beamLocal Oscillator Feedbackfilter Vacuum chamber Δ AOM CavityMembrane SpectrumanalyserFBTGTone
Fig. 1. (Color online) A 1064 nm laser generates two beams. Theprobe beam, indicated by blue lines, is used to lock the laser fre-quency to the cavity resonance. Its phase, in which the mem-brane mechanical motion is encoded, is monitored with a homodynescheme. The cooling beam, represented by red lines, provides theoptomechanical interaction and is enclosed within a feedback loop.After being transmitted by the cavity, its amplitude is detected andthe resulting signal is electronically processed and used to modulatethe amplitude of the input field. In this way both the noise propertiesof light and the cavity susceptibility are modified. fluctuations via balanced homodyne detection. The cooling(pump) beam, detuned by a frequency ∆ from the cavity res-onance by means of two acousto–optic modulators (shownschematically as AOM in Fig. 1), drives the optical cavity andprovides the optomechanical interaction. This field is not acoherent, free field, but is subjected to a feedback, i.e. it is anin–loop field. After being filtered by the cavity, the amplitudequadrature of the transmitted field is directly detected with asingle photodiode. The resulting photocurrent is amplified,filtered and fed back to the AOM driver in order to modulatethe amplitude of the input field, thus closing the loop. The fullcharacterisation of the feedback response function is reportedin Refs. [7, 8], where we have already demonstrated that thiskind of feedback can be employed to enhance the e ffi ciencyof optomechanical sideband cooling. In particular we haveshowed how the in-loop spectra change when the feedbackgoes from positive to negative.Enclosing the optical cavity within the loop [7–11, 16] ef-fectively modifies its susceptibility for the in–loop opticalfield, such that (see also [17])˜ χ e ff c ( ω ) = ˜ χ c ( ω )1 − ˜ χ fb ( ω ) (cid:2) ˜ χ c ( ω ) e − i θ ∆ + ˜ χ ∗ c ( − ω ) e i θ ∆ (cid:3) , (1)where ˜ χ c ( ω ) = [ κ + i( ω − ∆ )] − is the cavity susceptibility,˜ χ fb ( ω ) = η √ κ κ (cid:48) √ n s ˜ g fb ( ω ), with η the detection e ffi ciency, κ and κ (cid:48) the input and output cavity decay rate respectively, n s the mean intracavity photon number, and ˜ g fb ( ω ) the feed-back control function [˜ g ∗ fb ( − ω ) = ˜ g fb ( ω )]. Furthermore, thedimensionless displacement of the mechanical oscillator mea-sured by the out–of–loop probe beam, δ ˜ q = ˜ χ o , e ff m ( ω )[ ˜ ξ ( ω ) + (cid:101) N e ff ( ω )] [17], is the sum of a term proportional to thermalnoise, described by the zero mean stochastic noise operator ˜ ξ ( ω ), and a term due to the interaction with the cavity, pro-portional to radiation pressure noise, reshaped by the e ff ec-tive cavity susceptibility according to the relation (cid:101) N e ff ( ω ) = G { ˜ χ e ff c ( ω ) ˜ n + [ ˜ χ e ff c ( ω )] ∗ ˜ n † } , with ˜ n the radiation pressure noiseoperator [17] and G = g √ n s the (many–photon) optome-chanical coupling strength [2, 18], where g is the single–photon optomechanical coupling. Finally, in the expressionfor the mechanical displacement, the factor ˜ χ o , e ff m ( ω ) is themodified mechanical susceptibility that is dressed by the ef-fective self–energy Σ e ff ( ω ) = − i G { ˜ χ e ff c ( ω ) − [ ˜ χ e ff c ( − ω )] ∗ } ac-cording to [ ˜ χ o , e ff m ( ω )] − = [ ˜ χ m ( ω )] − + Σ e ff ( ω ) , (2)where the bare susceptibility is [ ˜ χ m ( ω )] − = ( ω − ω − i ωγ m ) /ω m .In the resolved sideband limit, ω m (cid:29) κ , and for ∆ ∼ ω m in order to cool the resonator, the e ff ective cavity suscepti-bility for frequencies close to the cavity resonance ω ∼ ∆ can be approximated as ˜ χ e ff c ( ω ) ∼ [ κ e ff + i( ∆ e ff − ω )] − , where κ e ff = κ + Im[ ˜ χ fb ( ∆ )] and ∆ e ff = ∆ − Re[ ˜ χ fb ( ∆ )]. These rela-tions allow to significantly simplify the expressions reportedabove and interpret the system dynamics in terms of that ofa standard optomechanical system with a modified cavity. Inparticular, in the positive feedback regime (corresponding tolight anti–squashing) the in–loop optical mode experiencesan e ff ectively reduced decay rate, which tends to zero as thefeedback gain is increased and approaches the feedback insta-bility [7, 8]. This in turn amounts to an increased optome-chanical cooperativity C e ff = G /κ e ff γ m . In Refs. [7, 8]we have correspondingly shown that this e ff ect can be em-ployed to augment the mechanical damping rate Γ e ff and henceto improve sideband cooling of mechanical motion. Herewe demonstrate that in–loop optical cavities represent a new,powerful tool for reaching the strong coupling regime, owingto an e ff ective reduction of the cavity linewidth κ e ff .Normal–mode splitting is a clear signature of strong cou-pling, being that it is only observable above the threshold G (cid:38) κ e ff [2, 4] (in typical optomechanical systems the othercondition G > γ m is easily satisfied). Since both normalmodes are combinations of light and mechanical modes, theyare both visible in the detectable mechanical displacementspectrum as distinct peaks at frequencies ω ± , separated by ω + − ω − (cid:39) √ G when ∆ e ff = ω m . The two peaks are dis-tinguishable if the corresponding linewidths, which are of theorder of κ e ff , are smaller than G . In particular, strong cou-pling manifests itself as avoided crossing for the values ofthe normal frequencies ω ± when the cavity detuning is var-ied. This is apparent from Fig. 2, showing the spectra of thedisplacement fluctuations of the mechanical mode interactingwith the in–loop optical mode, recorded via homodyne detec-tion of the probe beam. In Fig. 2a) a color–plot is used to showthese spectra as a function of frequency and normalised de-tuning, acquired with the maximum attainable feedback gain,and panel b) is the theoretical expectation. The parametersused for the simulation, determined independently, are the de-cay rate κ = π ×
22 kHz, the single–photon optomechanicalcoupling estimated to be g = π × . P = µ W. These pa-rameters correspond to G ∼ π × γ m , but lower than κ , implying that the optomechanical systemis initially far from the strong–coupling regime. The feedbackis then set to operate in the anti–squashing regime, with sucha value of gain that the threshold G ∼ κ e ff is surpassed andnormal mode splitting becomes visible.Let us now analyse these spectra in more detail. In theresolved sideband limit, the symmetrised displacement noisespectrum can be expressed as [17] S qq ( ω ) (cid:39) (cid:12)(cid:12)(cid:12) ˜ χ o , e ff m ( ω ) (cid:12)(cid:12)(cid:12) [ S th + S e ff rp ( ω ) + S fb ( ω )] , (3)where the first two terms account for the standard spectrum(with no feedback) for an optomechanical system, but withcavity decay rate κ e ff , and the last term can be interpreted asadditional noise due to the feedback and is given by [17] S fb ( ω ) ∼ G Z ∆ [ (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ ˜ χ e ff c ( − ω )] (cid:12)(cid:12)(cid:12) ] , (4)which has the same form of the radiation pressure term, ex-cept for the factor Z ∆ = [( ∆ − ∆ e ff ) + ( κ e ff − κ ) ] / ηκ (cid:48) replac-ing κ e ff . Fig. 3 shows the spectrum of the fundamental me-chanical mode excited by thermal fluctuations at 300 K (bluetrace), with an optomechanical contribution due to the quasi–resonant probe beam with 15 µ W of power, which slightlycools down the mechanical mode, increasing the damping rateby a factor of ∼ .
8, due to an estimated probe detuning ofaround 2 π ×
300 Hz. The red trace demonstrates the standard(no feedback) sideband–cooling due to the cooling beam witha detuning set to ∆ = π ×
330 kHz, and the other optomechan-ical parameters set as for the data in Fig. 2, such that the strongcoupling regime is initially not reached. Finally, the greentrace corresponds to the cross–section of Fig. 2a) indicated -27 -28 -29 -30 -31
340 345 350 340 345 350 a) b)
Fig. 2. (Color online) Normal mode splitting. a), Measured, and b),theoretically predicted splitting of the fundamental mechanical modein the strong-coupling regime as a function of detuning, with the twonormal modes exhibiting avoided crossing. The dashed grey lineindicates the optimal value of the detuning for sideband–cooling withfeedback. The values of the colour scale are in m / Hz and correspondto the displacement spectral noise evaluated as S xx ( ω ) = x S qq ( ω )with x = √ (cid:126) / m ω m the zero point motion factor, and S qq ( ω ) thepower spectrum of the dimensionless displacement operator δ q [17]. by the grey dashed line. In this particular case we estimate,from the experimental data and the simulation, the e ff ectiveparameters κ e ff ∼ π × ∆ e ff ∼ π × .
65 kHz.Since Z ∆ (cid:29) κ e ff in the range of parameters relevant to ourexperiment, the feedback noise, di ff erently from the radiationpressure term, provides a non-negligible contribution to theoverall spectrum with respect to the thermal one, as indicatedby the dashed and dotted lines.The results we have presented are obtained in a conditionin which the pump field e ffi ciently cools the mechanical res-onator [7, 8]. In general, when an optomechanical systementers the strong coupling regime, the e ffi ciency of sidebandcooling decreases. Hereafter we report on the similar e ff ectthat we observe as we increase the feedback gain towards in-stability, while keeping the other parameters fixed, as shownin Fig. 4. Panel a) presents a plot of the mechanical dis-placement spectra as a function of frequency and feedbackgain G fb = − Im[ ˜ χ fb ( ∆ )] /κ , normalised in such a way that G fb = κ e ff =
0, i.e. at the feedback stability thresh-old. In panel b) we report the corresponding and consistentresults simulated using the theoretical model with the previ-ously listed parameters for the membrane mode, P = µ Wand ∆ = π × . κ = π ×
21 kHzand g = π × . ffi ciency increases with the feedbackgain. As explained previously, this e ff ect can be understoodas a result of the increment in the optomechanical cooperativ-
335 340 345 350 ω/ π (kHz) − − − − − − S xx ( m / H z ) Fig. 3. (Color online) Displacement spectral noise, S xx ( ω ) = x S qq ( ω ) with x = √ (cid:126) / m ω m and o ff set by the shot–noise greytrace, of the (0,1) membrane mode at room temperature (blue trace),and sideband–cooled (red trace) with a pump of P = µ W detunedby ∆ =
330 kHz. Increasing the gain with the feedback operating inthe anti–squashing regime e ff ectively reduces the cavity linewidth,allowing to enter the strong–coupling regime, as seen from the ap-pearance of two hybrid modes (green trace). The green–solid linerepresents the theoretical expectation according to eq. (3), and is thesum of the comparable thermal and feedback terms shown as dotted–and dashed–line, respectively, while the radiation pressure contribu-tion is negligible. The narrow feature at ∼
339 kHz is a calibrationtone. -28 -29 -30 -31 a)b)c) Fig. 4. (Color online) Transition between the weak– and strong–coupling regime. a), Power spectra of mechanical displacement fluc-tuations varying the feedback gain G fb , and b), the correspondingsimulation evaluated as S xx ( ω ) = x S qq ( ω ) with x = √ (cid:126) / m ω m .The color scale is shown at the top in m / Hz. At low gain the me-chanical motion is described by a single mode the dynamics of whichis modified by the in–loop optomechanical interaction. At high gain,instead, the spectrum becomes double–peaked: the strong interactionproduces hybridized optomechanical modes and the mechanical mo-tion is a superposition of these two normal modes. The di ff erencein frequency between the two peaks in the spectrum with maximumgain corresponds to G = π × .
87 kHz. The parameters for thismeasurement ( P , ∆ , κ and g ) yield G = π × .
96 kHz, in goodagreement with the experimental estimation. c), The ratio of the ef-fective phonon number with and without feedback. The circles areobtained by numerical integration of the measured spectra, while thesolid line corresponds to Eq. (5) evaluated using the measured pa-rameters, which is valid both in the weak– and in the strong–couplingregime. The feedback scheme enhances the cooling rate with respectto standard sideband–cooling by a factor of 5 dB. ity due to the e ff ectively reduced in–loop cavity decay rate.We further note that, as expected, the enhanced cooperativitydoes not imply an improvement of optical cooling all the waytowards the instability point. Rather, the cooling works wellin the weak–coupling limit, i.e. when the cavity response time κ − ff is shorter than the decay time of the oscillator modified bythe optomechanical interaction Γ − ff , so as to allow mechani-cal thermal energy to be transferred into the cavity mode andleak out [18]. Conversely, around the threshold κ e ff ∼ G theoverall mechanical damping rate is of the order of the cav-ity linewidth, Γ e ff ∼ κ e ff , and as the gain is increased further, Γ e ff grows, while κ e ff gets smaller, such that the cooling e ffi - ciency decreases. In particular, for high temperature, in theresolved sideband limit, κ e ff (cid:28) ∆ e ff ∼ ω m , small optomechan-ical coupling G (cid:28) ω m , and a small mechanical decay rate γ m (cid:28) ( Γ e ff , κ e ff ), the steady state average number of mechani-cal excitations n m can be evaluated in terms of the integral ofthe spectrum S qq ( ω ) [18], and it is given by [17] n m ∼ n th , e ff m γ m Γ e ff (cid:32) + Γ e ff κ e ff (cid:33) , (5)which is equal to the result for a standard optomechanical sys-tem (with no feedback), but with cavity decay rate κ e ff , and ina higher temperature reservoir n th , e ff m ∼ n thm + n e ff m , with n e ff m ∼ Z ∆ Γ e ff γ m (2 κ e ff + Γ e ff ) . (6)The validity of this result is demonstrated in Fig. 4c) where wereport the e ff ective phonon number of the mechanical mode,normalised with respect to the occupancy obtained by stan-dard sideband–cooling without feedback, n SCm . In particular,the solid line, which is in very good agreement with the data(dots), represents the expected average phonon number de-fined in Eq. (5). The optimal cooling gain is G fb ≈ . ff ers the possibility to tune the e ff ective cavity linewidth atwill. In particular, herein we have shown that this allows toaccess the regime of strong coupling, characterised by theemergence of hybridized normal modes, even when the op-tomechanical interaction is small as compared to the naturaldissipation rates, so that the original system is in fact weaklycoupled. In our experiment, using the optimal parameters ofFig. 2, the e ff ective cavity decay rate is reduced by a factor20, and the system is promoted to the strong coupling regimewith an estimated cooperativity parameter of C e ff (cid:39) × .We further note that the ability to e ff ectively reduce the cav-ity linewidth may ease tasks such as transduction, storageand retrieval of signals and energy [19–21] with low fre-quency massive resonators. Finally, this technique could alsobe exploited to improve certain protocols for the preparationof non-classical mechanical states [22], which are more e ffi -cient at low cavity decay rate, or to enhance the e ffi ciency ofmechanical heat engines which work in the strong couplingregime [23] or which make use of correlated reservoirs [24]. Acknowledgments –
We acknowledge the support ofthe European Commission through the H2020-FETPROACT-2016 project n. 732894 “HOT”. ∗ Present address:
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Massimiliano Rossi,
1, 2, *
Nenad Kralj, Stefano Zippilli,
2, 3
Riccardo Natali,
2, 3
Antonio Borrielli, Gregory Padraud, Enrico Serra,
5, 6
Giovanni Di Giuseppe,
2, 3, † and David Vitali
2, 3, 7 ‡ School of Higher Studies “C. Urbani”, University of Camerino, 62032 Camerino (MC), Italy School of Science and Technology, Physics Division, University of Camerino, 62032 Camerino (MC), Italy INFN, Sezione di Perugia, 06123 Perugia (PG), Italy Institute of Materials for Electronics and Magnetism,Nanoscience-Trento-FBK Division, 38123 Povo (TN), Italy Delft University of Technology, Else Kooi Laboratory, 2628 Delft, The Netherlands Istituto Nazionale di Fisica Nucleare, TIFPA, 38123 Povo (TN), Italy CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy
THEORY
We consider a mode of a mechanical resonator described bythe dimensionless position and momentum operators q and p (with (cid:2) q , p (cid:3) = i) with frequency ω m , decay rate γ m , and mass m . The operators q and p are related to the real position ˆ x = x q and momentum ˆ P = p p , by the factors x = √ (cid:126) / m ω m and p = √ (cid:126) m ω m /
2. The mechanical resonator is coupled toa resonant mode of a Fabry–P´erot cavity, with operators a and a † ([ a , a † ] = ω c and with decay rate κ , viaradiation pressure with strength g = − x d ω c / dx . The cavityis driven by a laser field at frequency ω L , amplitude modulatedby a feedback system, which measures the light transmittedby the cavity (see Fig. S1). The equations of motion for thissystem are˙ q = ω m p , (S1)˙ p = − ω m q − γ m p + √ g a † a + ξ , (S2)˙ a = − ( κ + i ∆ ) a + i √ g aq + (cid:112) κ A in e − i θ ∆ ++ √ κ (cid:48) a (cid:48) in + √ κ (cid:48)(cid:48) a (cid:48)(cid:48) in , (S3)where ∆ = ω c − ω L is the beam detuning, and the phase θ ∆ = arctan( − ∆ /κ ) accounts for having chosen the phase ofthe cavity field as reference, with ∆ the e ff ective detuningdefined below. The operator ξ describes thermal noise act-ing on the mechanical resonator and, in the high temperaturelimit relevant here, is characterized by the correlation function (cid:104) ξ ( t ) ξ ( t (cid:48) ) (cid:105) = γ m (2 n th + δ ( t − t (cid:48) ) [S1], with n th the number ofthermal excitations. Moreover, we decompose the total cavitydecay rate as κ = κ + κ (cid:48) + κ (cid:48)(cid:48) , in terms of the contributions dueto the losses of the input mirror κ , output mirror κ (cid:48) and addi-tional internal losses κ (cid:48)(cid:48) . Correspondingly we have introducedthree input operators A in , a (cid:48) in and a (cid:48)(cid:48) in . In particular, the lattertwo describe vacuum noise and are characterized by the cor-relation function (cid:68) a (cid:48) in ( t ) a (cid:48) in † ( t (cid:48) ) (cid:69) = (cid:68) a (cid:48)(cid:48) in ( t ) a (cid:48)(cid:48) in † ( t (cid:48) ) (cid:69) = δ ( t − t (cid:48) ),instead the operator associated to the field at the input mirrorcan be decomposed as A in = a in + E + Φ , (S4) where a in describes vacuum noise, E = √P / (cid:126) ω L accounts forthe pump field at power P , and Φ is the contribution due tothe feedback. m AOM I ( t ) ( t ) a q Fig. S1. Sketch of the optomechanical system.
This last part can be expressed as Φ ( t ) = (cid:90) t −∞ d t (cid:48) g fb ( t − t (cid:48) ) I ( t (cid:48) ) , (S5)where g fb ( t ) is the causal filter function of the feedback and I ( t ) describes the detection photocurrent given by I ( t ) = (cid:104) √ η a † out + (cid:112) − η c † (cid:105) (cid:104) √ η a out + (cid:112) − η c (cid:105) = η a † out a out + (cid:112) η (1 − η ) ( a † out c + c † a out ) + (1 − η ) c † c , (S6)where η is the detection e ffi ciency, c represents additional vac-uum noise due to the ine ffi ciency of the detection, and a out describes the transmitted output field given by a out = √ κ (cid:48) a − a (cid:48) in . (S7)Approximate solutions of Eqs. (S1)–(S3) can be found, pro-vided the system is stable, by linearisation of the system forsmall fluctuations, δ a and δ q , around the steady state solution α s = a − δ a and q s = q − δ q . These values are determined byimposing (cid:104) ˙ q (cid:105) = (cid:104) ˙ a (cid:105) =
0, and are given by q s = √ g α ω m α s = √ κ | κ + i ∆ | ( E + ¯ Φ ) , (S8)where we have introduced the e ff ective detuning ∆ = ∆ −√ g q s = ∆ − g α /ω m , and the averaged feedback re-sponse ¯ Φ = ηκ (cid:48) α (cid:82) ∞ d tg fb ( t ).The linearised equations for the fluctuations (obtained byneglecting contributions at second order in the fluctuations)are δ ¨ q = − ω δ q − γ m δ ˙ q + ω m √ g α s ( δ a + δ a † ) + ω m ξ , (S9) δ ˙ a = − ( k + i ∆ ) δ a + i √ g α s δ q + (cid:112) κ δ Φ a e − i θ ∆ ++ (cid:112) κ ( δ Φ n + a in ) e − i θ ∆ ++ √ κ (cid:48) a (cid:48) in + √ κ (cid:48)(cid:48) a (cid:48)(cid:48) in , (S10)where we have eliminated the equation for p , and the feedbackresponse Φ has been decomposed as Φ = ¯ Φ + δ Φ a + δ Φ n with δ Φ a = ηκ (cid:48) α s (cid:90) t −∞ d t (cid:48) g fb ( t − t (cid:48) ) (cid:16) δ a ( t (cid:48) ) + δ a † ( t (cid:48) ) (cid:17) , (S11) δ Φ n = − η √ κ (cid:48) α s (cid:90) t −∞ d t (cid:48) g fb ( t − t (cid:48) ) (cid:16) a (cid:48) in ( t (cid:48) ) + a (cid:48) † in ( t (cid:48) ) (cid:17) ++ (cid:112) η (1 − η ) √ κ (cid:48) α s (cid:90) t −∞ d t (cid:48) g fb ( t − t (cid:48) ) (cid:16) c ( t (cid:48) ) + c † ( t (cid:48) ) (cid:17) . (S12)In the frequency domain, where operators are indicated by thetilde symbol ˜, we find − i ω δ ˜ a = − ( k + i ∆ ) δ ˜ a + i G δ ˜ q + (cid:112) κ δ ˜ Φ a e − i θ ∆ + ˜ n , − ω δ ˜ q = − ω δ ˜ q + i ω γ m δ ˜ q + G ω m ( δ ˜ a + δ ˜ a † ) + ω m ˜ ξ , (S13)where G = g √ n s is the optomechanical coupling with n s = α the mean intracavity photon number,˜ n = (cid:112) κ ( δ ˜ Φ n + ˜ a in )e − i θ ∆ + √ κ (cid:48) ˜ a (cid:48) in + √ κ (cid:48)(cid:48) ˜ a (cid:48)(cid:48) in , (S14)and δ ˜ Φ a = η κ (cid:48) α s ˜ g fb ( ω ) (cid:16) δ ˜ a + δ ˜ a † (cid:17) , (S15) δ ˜ Φ n = − η √ κ (cid:48) α s ˜ g fb ( ω ) (cid:16) ˜ a (cid:48) in + ˜ a (cid:48) † in (cid:17) ++ (cid:112) η (1 − η ) √ κ (cid:48) α s ˜ g fb ( ω ) (cid:16) ˜ c + ˜ c † (cid:17) , (S16)with ˜ g fb ( ω ) the Fourier transform of the filter function whichfulfils the relation ˜ g fb ( ω ) ∗ = ˜ g fb ( − ω ). We note that when con-sidering operators in Fourier space the symbol † is not usedto indicate the hermitian conjugate of the corresponding op-erator, rather, the hermitian conjugate of the operator at theopposite frequency, so that given a generic operator ˜ o ≡ ˜ o ( ω ),then ˜ o † ≡ [ ˜ o ( − ω )] † . System response functions
Let us now introduce some quantities that will be useful inthe following discussion: the bare cavity susceptibility˜ χ c ( ω ) = [ κ + i( ∆ − ω )] − , (S17) the bare mechanical susceptibility˜ χ m ( ω ) = ω m (cid:104) ω − ω − i ωγ m (cid:105) − , (S18)the e ff ective cavity susceptibility modified by the feedback˜ χ e ff c ( ω ) = ˜ χ c ( ω )1 − ˜ χ fb ( ω ) (cid:2) ˜ χ c ( ω ) e − i θ ∆ + ˜ χ ∗ c ( − ω ) e i θ ∆ (cid:3) , (S19)the e ff ective dressed mechanical susceptibility modified bythe optomechanical interaction and by the feedback˜ χ o , e ff m ( ω ) = (cid:110) ˜ χ m ( ω ) − + Σ e ff ( ω ) (cid:111) − , (S20)with Σ e ff ( ω ) = − i G (cid:104) ˜ χ e ff c ( ω ) − ˜ χ e ff c ( − ω ) ∗ (cid:105) , (S21)and the rescaled filter function˜ χ fb ( ω ) = η (cid:112) κ κ (cid:48) α s ˜ g fb ( ω ) . (S22)We finally note that, in the resolved sideband limit κ (cid:28) ω m and for short feedback delay time κ (cid:28) /τ fb , the e ff ective cav-ity susceptibility ˜ χ e ff c ( ω ) can be approximated, for frequenciesaround the cavity resonance ω ∼ ∆ , as the bare susceptibilityof a cavity with e ff ective cavity decay rate κ e ff and detuning ∆ e ff , such that [S2]˜ χ e ff c ( ω ) ∼ κ e ff + i( ∆ e ff − ω ) . (S23)The values of κ e ff and detuning ∆ e ff can be expressed in termsof the feedback filter function (which is slowly varying andessentially constant over the cavity linewidth) evaluated forfrequencies close to the cavity resonance ˜ χ fb ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) ω ∼ ∆ ≡ ˜ χ ∆ fb , as κ e ff = κ + Im (cid:104) ˜ χ ∆ fb (cid:105) and ∆ e ff = ∆ − Re (cid:104) ˜ χ ∆ fb (cid:105) , that is˜ χ ∆ fb ∼ ∆ − ∆ e ff + i ( κ e ff − κ ) . (S24) Power spectrum of the mechanical position operator
An expression for the displacement operator δ ˜ q can be de-rived solving Eq. (S13). It reads δ ˜ q = ˜ χ o , e ff m ( ω ) (cid:104) ˜ ξ ( ω ) + (cid:101) N e ff ( ω ) (cid:105) , (S25)where (cid:101) N e ff ( ω ) = G (cid:104) ˜ χ e ff c ( ω ) ˜ n + [ ˜ χ e ff c ( − ω )] ∗ ˜ n † (cid:105) . (S26)Using this expression it is possible to evaluate the power spec-trum of the mechanical displacement operator δ ˜ q ( ω ) that canbe measured by sending a probe field resonant with the cav-ity mode such that mechanical fluctuations modulate the fieldphase. From the measurement of the probe field phase itis possible to determine the displacement spectrum which isgiven by S qq ( ω ) = (cid:90) d ω (cid:48) (cid:2) (cid:104) δ ˜ q ( ω ) δ ˜ q ( ω (cid:48) ) (cid:105) + (cid:104) δ ˜ q ( − ω ) δ ˜ q ( ω (cid:48) ) (cid:105) (cid:3) . (S27)The expression for this spectrum can be decomposed in a sumof three terms as S qq ( ω ) = (cid:12)(cid:12)(cid:12) ˜ χ o , e ff m ( ω ) (cid:12)(cid:12)(cid:12) (cid:104) S th + S κ rp ( ω ) + S fbrp ( ω ) (cid:105) , (S28)where the first is due to thermal noise and the other two aredue to radiation pressure noise. They are explicitly given by S th = γ m (cid:16) n thm + (cid:17) , S κ rp ( ω ) = G κ (cid:20)(cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( − ω ) (cid:12)(cid:12)(cid:12) (cid:21) , S fbrp ( ω ) = G | ˜ χ fb ( ω ) | η κ (cid:48) (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) e − i θ ∆ + [ ˜ χ e ff c ( − ω )] ∗ e i θ ∆ (cid:12)(cid:12)(cid:12) − G χ fb ( ω )] ∗ (cid:110) ˜ χ e ff c ( ω ) + [ ˜ χ e ff c ( − ω )] ∗ (cid:111) × (cid:110) ˜ χ e ff c ( − ω ) e − i θ ∆ + [ ˜ χ e ff c ( ω )] ∗ e i θ ∆ (cid:111) − G χ fb ( ω ) (cid:110) ˜ χ e ff c ( − ω ) + [ ˜ χ e ff c ( ω )] ∗ (cid:111) × (cid:110) ˜ χ e ff c ( ω ) e − i θ ∆ + [ ˜ χ e ff c ( − ω )] ∗ e i θ ∆ (cid:111) . (S29)where S κ rp ( ω ) is proportional to the standard radiation pressureterm in a cavity with modified susceptibility ˜ χ e ff c ( ω ), while theterm S fbrp ( ω ) is given by the noise term in the feedback re-sponse function δ ˜ Φ n [see Eqs. (S14) and (S16)].Let us consider the term S fbrp ( ω ) more closely. Since weoperate the feedback close to instability, where the e ff ectivecavity susceptibility ˜ χ e ff c ( ω ) has a very narrow linewidth κ e ff ,so that it is relevant only for a relatively narrow frequencyrange around the cavity resonance (i.e. it is peaked at ω =∆ e ff over a bandwidth of the order of κ e ff (cid:28) ∆ e ff ), we canapproximate ˜ χ e ff c ( ω ) ˜ χ e ff c ( − ω ) ∼
0, and in turn write S fbrp ( ω ) as S fbrp ( ω ) (cid:39) G (cid:20) Z ( ω ) (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) + Z ( − ω ) (cid:12)(cid:12)(cid:12) [ ˜ χ e ff c ( − ω )] (cid:12)(cid:12)(cid:12) (cid:21) (S30)with Z ( ω ) = | ˜ χ fb ( ω ) | η κ (cid:48) − Re (cid:104) ˜ χ fb ( ω ) e − i θ ∆ (cid:105) , (S31)which changes slowly over the range of frequencies aroundthe mechanical mode, and can be approximated by its valueclose to the cavity resonance Z ( ω ) ∼ Z ( ω ) | ω ∼ ∆ . Accordingto Eq. (S24) it can be expressed in terms of the e ff ective cavitydecay rate and detuning.It is useful to decompose the position spectrum as S qq ( ω ) (cid:39) (cid:12)(cid:12)(cid:12) ˜ χ o , e ff m ( ω ) (cid:12)(cid:12)(cid:12) (cid:104) S th + S e ff rp ( ω ) + S fb ( ω ) (cid:105) , (S32)where the first two terms account for the standard spectrum(with no feedback) for an optomechanical system, but withcavity decay rate κ e ff , such that S e ff rp ( ω ) = G κ e ff (cid:20)(cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( − ω ) (cid:12)(cid:12)(cid:12) (cid:21) . (S33) The last term can be interpreted as additional noise due to thefeedback S fb ( ω ) = S fbrp ( ω ) + S κ rp ( ω ) − S e ff rp ( ω ) = G (cid:20) Z ( ω ) (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) + Z ( − ω ) (cid:12)(cid:12)(cid:12) [ ˜ χ e ff c ( − ω )] (cid:12)(cid:12)(cid:12) (cid:21) , (S34)with Z ( ω ) = Z ( ω ) + κ − κ e ff , (S35)which changes slowly over the range of frequencies of inter-est, and can be approximated with its value at the cavity fre-quency Z ( ω ) | ω ∼ ∆ ≡ Z ∆ (see Eq. (S24)) as Z ( ω ) ∼ Z ∆ = ( ∆ − ∆ e ff ) + ( κ e ff − κ ) η κ (cid:48) . Correspondingly S fb ( ω ) ∼ G Z ∆ (cid:20)(cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ ˜ χ e ff c ( − ω )] (cid:12)(cid:12)(cid:12) (cid:21) , (S36)which has the same form of the radiation pressure term, ex-cept for the factor Z ∆ replacing κ e ff . Since Z ∆ (cid:29) κ e ff in theparameter range relevant to our experiment, this latter feed-back noise provides a non-negligible contribution to the over-all spectrum (see Fig. 3 of the main text) as opposed to radia-tion pressure. Steady state phonon number
In the resolved sideband limit, that is small cavity decayrate κ e ff (cid:28) ∆ e ff ∼ ω m and small optomechanical coupling G (cid:28) ω m , the steady state average number of mechanical ex-citations n m can be evaluated in terms of the integral of thespectrum S qq ( ω ) [S3]: n m + / (cid:39) (cid:68) δ q (cid:69) (cid:39) (cid:68) δ p (cid:69) (cid:39) π (cid:90) ∞−∞ d ω S qq ( ω ) . (S37)The integral can be computed by means of the analytical ex-pressions reported in [S3]. In order to be able to apply thoseformulas in the present case we have to consider another de-composition of the position spectrum. Specifically, we findthat S qq ( ω ) is proportional to the spectrum for a standard op-tomechanical system (with no feedback) with cavity decayrate κ e ff and a modified temperature, that is S qq ( ω ) = S (cid:48) qq ( ω ) κ e ff + Z ∆ κ e ff , (S38)where S (cid:48) qq ( ω ) ∼ (cid:12)(cid:12)(cid:12) ˜ χ o , e ff m ( ω ) (cid:12)(cid:12)(cid:12) (cid:104) S (cid:48) th + S e ff rp ( ω ) (cid:105) , (S39)with S (cid:48) th = S th κ e ff κ e ff + Z ∆ = γ m (cid:0) n (cid:48) m + (cid:1) , (S40)and n (cid:48) m = n thm κ e ff − Z ∆ (cid:2) κ e ff + Z ∆ (cid:3) . (S41)Now the integral of S (cid:48) qq ( ω ) is readily obtained by straightfor-ward application of the formulas reported in [S3]. Hence wefind, in the limit of small cavity decay rate κ e ff (cid:28) ∆ e ff ∼ ω m and when G (cid:28) ω m , that (cid:68) δ q (cid:69) ∼ κ e ff + Z ∆ κ e ff ( γ m + Γ e ff ) (cid:34) A + + A − + γ m n (cid:48) m (cid:32) + Γ e ff κ e ff (cid:33)(cid:35) , (S42)with A ± = G κ e ff κ ff + ( ∆ e ff ± ω m ) , (S43) Γ e ff = A − − A + . (S44)In particular, for high temperature, small mechanical decayrate γ m (cid:28) ( Γ e ff , κ e ff ), and resolved sideband limit such that Γ e ff ∼ A − (cid:29) A + , as in our case, the average number of me-chanical excitations is n m ∼ n th , e ff m γ m Γ e ff (cid:32) + Γ e ff κ e ff (cid:33) , (S45)which is equal to the result for a standard optomechanical sys-tem (with no feedback), but with cavity decay rate κ e ff , and ina higher temperature reservoir n th , e ff m ∼ n thm + n e ff m , (S46)with n e ff m = Z ∆ Γ e ff γ m (2 κ e ff + Γ e ff ) . (S47) Optomechanically induced transparency in the presence offeedback
We now focus on the combined e ff ect of the mechanicalresonator and of the feedback loop on cavity transmission. Tobe more specific, we consider the response of the system toan additional seed field injected from the input mirror, and westudy how it is transmitted through the cavity. In this way wecan study the e ff ect of feedback-controlled light on optome-chanically induced transparency [S4–S6].We are interested in the spectrum of the cavity amplitudefluctuations δ ˜ a + δ ˜ a † at the seed frequency, which is measuredby direct photodetection of the output field a out when the seedamplitude is much smaller than that of the pump, but still suf-ficiently large for the e ff ect of the input noise operators to benegligible. In fact, we find (neglecting terms at second orderin the field fluctuations) a † out a out ∼ κ (cid:48) (cid:104) α s + α s (cid:16) δ a + δ a † (cid:17)(cid:105) − √ κ (cid:48) α s (cid:16) a (cid:48) in + a (cid:48) † in (cid:17) . (S48) From Eq. (S13) we find that the amplitude fluctuations of thecavity field are described by the operator δ ˜ a + δ ˜ a † = ˜ χ o , e ff m ( ω )˜ χ m ( ω ) (cid:110) ˜ χ e ff c ( ω ) ˜ n + [ ˜ χ e ff c ( − ω )] ∗ ˜ n † + i G (cid:104) ˜ χ e ff c ( ω ) − [ ˜ χ e ff c ( − ω )] ∗ (cid:105) ˜ χ m ( ω ) ˜ ξ. (cid:111) (S49)In order to compute the transmission of the seed field we caninclude the seed amplitude at frequency ν in the noise operator˜ n → ˜ n + √ κ e − i θ ∆ α seed δ ( ω − ν ). Thereby we find that,neglecting all the noise terms, the transmitted field close tothe cavity resonance ( ν ∼ ∆ ) is˜ a † out ˜ a out ∼ κ (cid:48) α s (cid:16) δ ˜ a + δ ˜ a † (cid:17) ∼ ˜ t ( ν ) α seed , (S50)with the transmission coe ffi cient given by˜ t ( ω ) = κ (cid:48) (cid:112) κ e − i θ ∆ ˜ χ e ff c ( ω ) ˜ χ o , e ff m ( ω )˜ χ m ( ω ) , (S51)where ˜ χ o , e ff m ( ω )˜ χ m ( ω ) ∼ [ ˜ χ m ( ω )] − [ ˜ χ m ( ω )] − − i G ˜ χ e ff c ( ω ) = ω − ω − i ω γ m ω − ω − i ω γ m − i ω m G ˜ χ e ff c ( ω ) . (S52)The transmission spectrum is then given by S t ( ω ) = (cid:12)(cid:12)(cid:12) ˜ t ( ω ) (cid:12)(cid:12)(cid:12) = κ κ (cid:48) (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ χ o , e ff m ( ω )˜ χ m ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (S53)and can be expressed as a standard Fano profile [S7, S8],which describes interference phenomena, as S t ( ω ) = κ κ (cid:48) (cid:12)(cid:12)(cid:12) ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) (cid:34) ( (cid:15) + q ) (cid:15) + + ρ (cid:35) , (S54)where (cid:15) = ω − ω − ω m G Im (cid:104) ˜ χ e ff c ( ω ) (cid:105) γ m ω + ω m G Re (cid:104) ˜ χ e ff c ( ω ) (cid:105) , q = ω m G Im (cid:104) ˜ χ e ff c ( ω ) (cid:105) γ m ω + ω m G Re (cid:104) ˜ χ e ff c ( ω ) (cid:105) ,ρ = γ ω (cid:12)(cid:12)(cid:12) ω − ω − i ω γ m − i ω m G ˜ χ e ff c ( ω ) (cid:12)(cid:12)(cid:12) . (S55)Notice from Eq. (S54) that the Fano profile is determinedby the e ff ective cavity susceptibility modified by the feedbackloop, ˜ χ e ff c ( ω ). Destructive interference is observed when (cid:15) = − q , that is when ω = ω m , and ρ represents an additional smallterm proportional to the mechanical damping rate γ m whichprevents perfect destructive interference. When q = ω = ω m (that is when Im[ ˜ χ e ff c ( ω m )] = ∆ e ff = ω m ), the spectrum is symmetric with a dip in themiddle. Instead, the spectrum is asymmetric when q (cid:44) ω = ω m . The behaviour of the OMIT, the same as in standardoptomechanical systems, but with the cavity response modi-fied by the feedback, is experimentally verified in Fig. S2.
330 335 340 345 350 355 360 (a)
330 340 350 360 -1 (b)(c) Fig. S2. Optomechanically induced transparency (OMIT) in the pres-ence of feedback. (a) Color plot of the modulus square of the cavitytransmission | t ( ω ) | , as a function of frequency (horizontal scale) andfeedback gain normalized as defined in the Letter (vertical scale).The measurement is performed by injecting a seed on the in-loopcavity mode. The value of | t ( ω ) | is increasing from blue to red.At the mechanical resonance frequency, the seed is no more trans-mitted. The interference between the seed and the sideband createdby the mechanical mode on the red–detuned beam determines a de-structive interference, that is the OMIT phenomenon. Its width isdetermined by the optomechanical coupling, which is fixed, whilethe cavity decay rate κ e ff is modified by the feedback loop and de-creases for increasing feedback gain towards the instability. In (b)and (c) we plot, respectively, the magnitude square and phase of thecavity transmission for di ff erent fixed feedback gain increasing fromdark to light red. The black trace is obtained with no feedback. FINE–GAIN ATTENUATOR CALIBRATION
To explore deeply the instability region, and to reach thenormal–mode splitting regime, we have realised a circuit fora fine tuning of the feedback gain. An overall attenuationof 1 dB is divided in ten steps, each determined by threeappropriate resistors in Pi–configuration. The calibration ofthe feedback gain steps, which are used for the evaluationof the feedback gain in the experimental analysis, is reportedin Fig. S3.
Fig. S3. Calibration of the fine gain attenuator. ∗ Present address:
Niels Bohr Institute, University of Copenhagen,Blegdamsvej 17, 2100 Copenhagen, Denmark † [email protected] ‡ [email protected][S1] V. Giovannetti, and D. Vitali, Phys. Rev. A , 023812 (2001).[S2] M. Rossi et al. , Phys. Rev. Lett. , 123603 (2017).[S3] C. Genes et al. Phys. Rev. A , 033804 (2008).[S4] S. Weis, et al. , Science , 1520 (2010).[S5] A. H. Safavi-Naeini, et al. , Nature (London), , 69 (2011).[S6] M. Karuza et al. , Phys. Rev. A , 013804 (2013).[S7] U. Fano, Phys. Rev. , 1866 (1961).[S8] B. Lounis and C. Cohen-Tannoudji,
Journal de Physique II2