””Membrane-outside” as an optomechanical system

A. K. Tagantsev

1, 2, ∗ and E. S. Polzik Swiss Federal Institute of Technology (EPFL), School of Engineering,Institute of Materials Science, CH-1015 Lausanne, Switzerland Ioﬀe Phys.-Tech. Institute, 26 Politekhnicheskaya, 194021, St.-Petersburg, Russia Niels Bohr Institute, Quantum Optics Laboratory - QUANTOP,Blegdamsvej 17, DK-2100 Copenhagen, Denmark (Dated: February 24, 2021)

Abstract

We theoretically study an optomechanical system, which consists of a two-sided cavity and amechanical membrane that is placed outside of it. The membrane is positioned close to one of itsmirrors, and the cavity is coupled to the external light ﬁeld through the other mirror. Our studyis focused on the regime where the dispersive optomechanical coupling in the system vanishes.Such a regime is found to be possible if the membrane is less reﬂecting than the adjacent mirror,yielding a potentially very strong dissipative optomechanical coupling. Speciﬁcally, if the absolutevalues of amplitude transmission coeﬃcients of the membrane and the mirror, t and t m respectively,obey the condition t m < t (cid:28) t m (cid:28)

1, the dissipative coupling constant of the setup exceeds the dispersive coupling constant for an optomechanical cavity of the same length. The dissipativecoupling constant and the corresponding optomechanical cooperativity of the proposed system arealso compared with those of the Michelson-Sagnac interferometer and the so-called ”membrane-at-the-edge” system, which are known for a strong optomechanical dissipative interaction. It isshown that under the above condition, the system proposed here is advantageous in both aspects.It also enables an eﬃcient realization of the two-port conﬁguration, which was recently proposedas a promising optomechanical system, providing, among other beneﬁts, a possibility of quantumlimited optomechanical measurements in a system, which does not suﬀer from any optomechanicalinstability.

PACS numbers: 42.50.Lc, 42.50.Wk, 07.10.Cm, 42.50.Ct a r X i v : . [ qu a n t - ph ] F e b . INTRODUCTION Cavity quantum optomechanics is a promising branch of quantum optics. It allows forexploration of fundamental issues of quantum mechanics and paves a way for numerous ap-plications, e.g. in high-precision metrology and gravitational-wave defection . Mainly, thecavity optomechanics proﬁts from the so-called dispersive coupling , which originates fromthe dependence of the cavity resonance frequency on the position of a mechanical oscilla-tor. However, about a decade ago, Elste et al pointed out that the dispersive couplingdoes not provide the complete description of the optomechanical interaction. To ﬁll thegap, those authors have introduced the so-called dissipative coupling , which originates fromthe dependence of the cavity decay rate on the position of the mechanical oscillator. Sincethen such a coupling has been attracting an appreciable attention of theorists and ex-perimentalists . The dissipative coupling can do virtually all the jobs of the dispersivecoupling, such as optomechanical cooling, optical squeezing, and mechanical sensing whilethe physical conditions and mechanisms encountered in it are rather diﬀerent. Among the-oretical predictions analyzed for dissipative-coupling-assisted systems are the possibility ofsimultaneous squeezing and sideband cooling , a stable optical-spring eﬀect, which is not-feed-back-assisted , a virtually perfect squeezing of the optical noise in a system exhibitingno optomechanical instability , and not-feed-back-assisted cooling of a mechanical oscillatorunder the resonance excitation , the latter also demonstrated experimentally .The experimental implementations of the dissipative coupling are lagging signiﬁcantlybehind the theory. To a great extend this is due to the fact that the dissipative couplingis typically weak compared to the dispersive coupling. Hence even under speciﬁc tuningconditions when the dispersive coupling vanishes, it is typically diﬃcult to make the dis-sipative coupling eﬃcient. To date, the Michelson-Sagnac interferometer (MSI) has beentheoretically identiﬁed and experimentally addressed as a system, which can be tunedto be dominated by an ”anomalously strong dissipative coupling”. Recently, the so-called”membrane-at-the-edge” system (MATE), consisting of a one-sided cavity with a mem-brane placed inside the main resonator close to the input mirror, has been proposed as acandidate for an enhanced dissipative coupling. Another system dealing with an anoma-lously strong dissipative coupling is the popular ”membrane-in-the-middle” cavity onceit is driven close to the point of the spontaneous symmetry breaking . Here even diver-2 IG. 1. Membrane-outside system: a two-sided cavity with a membrane set behind the second(”non-feeding”) mirror. Dashed line - membrane, thin lines -semitransparent mirrors, thick arrows- pumping light, thin arrow - detected light. gence of dissipative coupling constants of individuals modes has been predicted. However,this system is characterized by tight doublets of modes with opposite signs of the dissipa-tive coupling constants, leading to cancelation of such divergencies and an optomechanicalperformance very diﬀerent from that of dissipative coupling of a single mode.In the present paper, we treat theoretically an optomechanical system which consists ofa two-sided cavity and a membrane that is placed outside of it, close to one of its mirrors.The cavity is coupled to the external laser light through the other mirror, and the lightleaving the cavity through that mirror is detected (Fig.1). We show that, for a properlypositioned membrane which is less reﬂecting than the adjacent cavity mirror, the dissipativecoupling in the system vanishes while the dissipative couling becomes anomalously strong.We have identiﬁed the interval of the membrane transparency providing the superior dissi-pative optomechanical coupling in this system which we refer to as a ”membrane-outside”(MOS).In addition to an example of an optomechanical device fully controlled by a strong dis-sipative coupling, MOS enables an eﬃcient realization of the two-port conﬁguration, whichwas recently proposed as a promising optomechanical system.We present a theoretical analysis of MOS (Sec.II), paying special attention to the two-port conﬁguration (Sect.III). A detailed comparison with MSI (Sec.IV) and MATE (Sec.V)in terms of the dissipative coupling constant and optomechanical cooperativity in the regimes3ominated by the dissipative coupling is provided. II. THEORETICAL ANALYSIS OF MOS

Primarily, we are interested in ﬁnding settings of MOS, under which it does not exhibitthe dispersive optomechanical coupling, and in evaluating the strength of the dissipativecoupling under these settings. A simple way to do this is to use the ”eﬀective mirror”approach (see, e.g. ) following which the tandem mirror/membrane (Fig.1) will be treatedas a synthetic mirror. For a ﬁxed cavity length l any variation of the mechanical variable,which is the distance x between the membrane and the mirror, will not aﬀect the cavityoptical length while the cavity decay rate and resonance frequency will be fully conditionedby the x -dependence of the power transmission coeﬃcient of the synthetic mirror and thatof the phase of its reﬂection coeﬃcient, respectively.We introduce the scattering matrices for the mirror it − r − r it (1)and for the membrane t m e iϕ t r m e iϕ r r m e iϕ r t m e iϕ t , (2)where t and r are the absolute values of the amplitude transmission and reﬂection coeﬃcients,respectively, which obeys the following relations t + r = 1 , t m + r m = 1 , and e i ( ϕ r − ϕ t ) = − . (3)Straightforward calculations (see Appendix A) yield T = t t m r r m + 2 rr m cos ψ , (4)for the power transmission coeﬃcient of the synthetic mirror andtan µ = r m t sin ψr m (1 + r ) cos ψ + r (1 + r m ) (5)for the phase of its the reﬂection coeﬃcient µ . Here ψ = 2 kx + ϕ r , (6)4here k is the light wave vector.The membrane position where the dispersive coupling vanishes is given by the condition d tan µdψ = 0 . (7)Such a derivative reads d tan µdψ = r m t r m (1 + r ) + r (1 + r m ) cos ψ [ r m (1 + r ) cos ψ + r (1 + r m )] . (8)Thus, as follows from Eqs.(7) and (8), the positions of the membrane where the dispersivecoupling vanishes should satisfy the following conditioncos ψ = − r m (1 + r ) r (1 + r m ) . (9)This condition can be met if r m < r, (10)i.e. the membrane should be less reﬂective than the adjacent mirror. Thus, under such acondition, at certain values of ψ , which is controlled by the position of membrane x , thesystem will be purely governed by the dissipative coupling, the situation we are looking for.Let us check if the positions given by Eq.(9) are of practical interest for implementationin optomechanics. For this purpose, we will ﬁnd the range of parameters of the syntheticmirror where (9) is compatible with the basic requirement T (cid:28) . (11)(if any) and evaluate the dissipative coupling in this compatible regime.Inserting (9) into (4), one ﬁnds T = t r m − r r m . (12)One readily checks that (10) ensures T ≤ t (cid:28) t m . (13)Now the strength of the dissipative coupling at the point where the dispersive couplingvanishes can be evaluated. First, if we neglect the energy stored in the synthetic mirror,5hich is a good approximation for x (cid:28) l (see Appendix B), the decay rate associated withthe synthetic mirror can be written as follows γ = cT l (14)where c is the speed of light. Thus, in view of (6), we ﬁnd dγdx = ckl dTdψ . (15)Next, the following relation dTdψ = 2 rr m t t m sin ψ [1 + r r m + 2 rr m cos ψ ] (16)and condition (9) yield (cid:12)(cid:12)(cid:12)(cid:12) dγdx (cid:12)(cid:12)(cid:12)(cid:12) = ckl t t m r m (1 + r m )1 − r m r (cid:115) r − r m − r m r ≈ ckl t t m r m (1 + r m ) , (17)where condition (13) was taken into account. This equation predicts a potentially verystrong dissipative coupling for t m < t .We conclude that, for the amplitude transparency of the membrane satisfying the condi-tion t m < t (cid:28) t m (cid:28) dissipative coupling, which is stronger than the dispersive coupling ckl for an optomechanical cavity of the same length.Equations (9), (18), and (4) imply that of interest are the positions of the membranewith x close to ˜ x = λ (cid:18)

12 + N + ϕ r π (cid:19) , λ = 2 π/k, (19)where cos ψ = − k ( x − ˜ x ) (cid:28) . (20)Keeping the lowest order terms in Φ, one readily ﬁnds a set of expressions that describe thesystem: T = t Φ Φ + Φ , (21)6 Td Φ = − t Φ Φ(Φ + Φ ) , (22)and dµd Φ = t − Φ (Φ + Φ ) , (23)where Φ = t m . (24)The above relations enable us to write simple explicit expressions for the cavity decayrate associated with the synthetic mirror γ = γ

11 + Φ / Φ , γ = 2 cl t t m (25)as well as for the optomechanical coupling constants, which we deﬁne as follows g ω = − dω c dx and g γ = − . dγdx , (26)where ω c is a resonance frequency of the system. Thus, keeping in mind that we are typicallyinterested in k that is very close to ω c /c , we can write g ω = g − Φ / Φ (1 + Φ / Φ ) , g γ = g / Φ (1 + Φ / Φ ) , g = 4 ω c l t t m . (27)Here, when calculating g ω , we use an approximate relation dω c dx = − c l dµdx (28)written neglecting the frequency dependence of µ , which is a good approximation for x (cid:28) l (see Appendix B).The optomechanical constants of the system and the decay rate associated with thesynthetic mirror, which are plotted as functions of the mirror position, are shown in Fig. 2and Fig. 3, respectively. As illustrated in Fig. 2, the position of the reﬂecting membranewith respect to the adjacent mirrow at which the dissipative coupling is much larger thanthe dispersive coupling is deﬁned by Φ / Φ ≈

1. This condition imposes the requirement onthe membrane position x − ˜ x ≈ λt m / π ≈ . t m = 10 − , λ = 0 . µ .It is also worth elucidating the origin of the enhancement of (cid:12)(cid:12) dγdx (cid:12)(cid:12) at decreasing t m . Asit is clear from Eqs.(20),(21), and (24), for t m (cid:28) x close to ˜ x , the transparency ofthe synthetic mirror exhibits a sharp maximum, cf., Fig.3. Its height scales as 1 /t m while7 IG. 2. Normalized dispersive (1) and dissipative (2) optomechanical constant of MOS, which areplotted as functions of Φ / Φ = 4( x − ˜ x ) k/t m , where x is the membrane position. g = ckl t t m , k is the wave vector of the light wave, and t m and t are the power transmission coeﬃcients of themembrane and mirror, respectively. ˜ x is given by Eq.(19). its width as t m , implying the average slope ∝ /t m . It is this t m -dependence that is seen inEq.(17) for (cid:12)(cid:12) dγdx (cid:12)(cid:12) .To conclude this Section, we would like to note that strictly speaking, the validity of thepresented above results may require a more stringent condition than x (cid:28) l . Speciﬁcally, asshown in Appendix B, the exact condition reads x (cid:28) l t m t . (29) III. IMPLICATION FOR SYMMETRIC TWO-SIDED CAVITY

It was recently shown that a two-port cavity, which is pumped through one of themirrors while the transparency of the other composite mirror is modulated with the mo-tion of a mechanical oscillator, is an optomechanical device that is promising for quantumstate generation and measurement, not suﬀering from any optomechanical instability. Forexample, it may be used for quantum limited measurements of the oscillator position and/or8 IG. 3. Normalized decay rate associated with the synthetic mirror, which is plotted as a functionof Φ / Φ = 4( x − ˜ x ) k/t m , where x is the membrane position and k is the wave vector of the lightwave. γ = cl t t m and t m and t are the power transmission coeﬃcients of the membrane andmirror, respectively. ˜ x is given by Eq.(19). for a virtually perfect light squeezing. This can be realized under the following conditions:(1) Resonance excitation, (2) Unresolved side-band regime (”bad cavity limit”), (3) Thesystem is dominated by the dissipative optomechanical coupling associated with the secondport, (4) The average transparency of the second mirror equals to that of the input mirror(”symmetric” cavity), (5) The output signal is that reﬂected from the input mirror. On theother hand, if a symmetric two-sided optomechanical cavity is dominated by the dispersivecoupling, the quantum limit cannot be reached such that only a 3dB squeezing in possible .MOS readily enables the realisation of such a device. For this purpose, one ﬁxes Φ = Φ by setting the membrane at the distance δx = λ t m π (30)from the position ˜ x given by Eq.(19) where the synthetic mirror is the most transparent.Under those conditions the system is governed by the dissipative coupling (Fig.2) and thedecay rate due to the synthetic mirros is equal to γ / γ / x = ˜ x , i.e. Φ = 0, the optomechanical coupling is purely dispersivewhile after a displacement of the membrane by δx , given by Eq.(30), it becomes purelydissipative. A transition between dissipative and dispersive types of coupling is of specialinterest since, according to Ref. 13, it corresponds to the transition between the states of thesystem where quantum limited measurements are possible and impossible, respectively. Foran ideal situation, where the intracavity losses are absent, such a transition is illustratedby Fig. 2b in Ref. 13. To estimate the eﬀect of the losses we follow Ref. 13 where thebackaction-imprecision product S imp xx S F F was calculated as a function of the ratio of thedispersive to the dissipative coupling constant. Here S imp xx is the equivalent displacementnoise power spectral density in the detected light (Fig.1) and S F F is the spectral densityof the quantum backaction force acting on the membrane. The cavity is driven with astrong monochromatic light. A quantum limited measurement is possible if S imp xx S F F = (cid:126) / (cid:126) is the Plank constant. We generalise the calculations of Ref. 13 by incorporatingan additional noise source characterized with the decay rate γ (see Appendix C) to ﬁnd S imp xx S F F = (cid:126) A + 2 Aξ ξ , ξ = g ω g γ , A = 1 + γ γ (31)for the resonance excitation of the symmetric cavity (the decay rate of both the syntheticand input mirror equals γ ). Equation (31) is plotted in Fig.4 for γ = 0 (”0 %”), γ = 0 . γ (”50%”), and γ = γ (”100%”). A clear persistence of the kink in this ﬁgure suggests thatthe ”switching” eﬀect in question is rather robust to the presence of the intracavity loss.The kink shown in Fig.4 provides a qualitative description of what happens when themembrane is shifted from a position with Φ = Φ to that with Φ = 0. Quantitatively, thekink is larger because, as follows from Fig.3, the shift from Φ = Φ to Φ = 0 also leadsto an increase of the decay rate associated with the synthetic mirror, which results in anadditional increase of the backaction-imprecision product in the dispersive limit.Note that the membrane-outside system considered here allows to achieve substantialquantum opto-mechanical cooperativity for the single photon ﬁeld circulating in the cav-ity. Consider the device pumped with a strong monochromatic light ( a is the number-of-photons-normalized amplitude of the pumping ﬁeld inside the cavity). Using the results10 .05 0.10 0.50 1 5 101.01.52.02.53.0 g ω / g γ S xx i m p S FF / ( ℏ / ) % % % FIG. 4. The impact of the intracavity losses on backaction-imprecision product for the symmetricmembrane-outside system. Normalized backaction-imprecision product plotted as a function ofthe ratio of the optomechanical constants g ω /g γ for γ = 0 (”0 %”), γ = 0 . γ (”50%”), and γ = γ (”100%”) where γ is the decay rate through each of the mirrors of the symmetric cavity, and γ is the decay rate associated with the intracavity losses. We assume the resonance excitation,unresolved sideband regime, and the average transmission of the synthetic mirror being equal tothat of the input mirror. from Ref. 13 for a symmetric two-sided MOS controlled by the dissipative coupling associ-ated with the ”non-feeding” mirror, the cooperativity, via (35) and (25), can be expressedas follows C = ( g γ x zpf a ) γγ m = M t t m , (32) M ≡ c ( ka x zpf ) lγ m , (33)where x zpf is the amplitude of zero-point ﬂuctuations and γ m is the decay rate of the mechan-ical oscillator. For state-of-the-art phononic bandgap membranes the amplitude of zero-point ﬂuctuations is x zpf = 10 − m and the mechanical decay rate is γ m = 0 . s − . With thecavity length l = 0 . t m = 0 . andt = 0 . IG. 5. Schematic of Michelson-Sagnac interferometer. The part marked with dashed-line rectan-gular can be considered as an eﬀective input mirror with x -dependent parameters such that thesystem can be viewed as a one-sided cavity . for the membrane and the adjacent mirror, respectively, we obtain close to unity cooperativ-ity for a single photon in the cavity ( a = 1). The symmetric cavity condition requires thatthe power transmission coeﬃcient of the input mirror is equal to the eﬀective power trans-mission coeﬃcient of the synthetic mirror T = 2 t /t m = 0 .

04. The corresponding ﬁnesse ofsuch symmetric cavity is F = π/T = 80. IV. COMPARISON WITH MICHELSON-SAGNAC INTERFEROMETER

The signal-recycled Michelson-Sagnac interferometer (MSI) is schematically depictedin Fig.5. It consists of three mirrors, a beam splitter, and a membrane shown with a wiggledline. This system can be viewed as a one-sided optomechanical cavity with an eﬀective12nput mirror, the parameters of which are functions of the membrane position . For certainmembrane positions the system is controlled exclusively by the dissipative coupling . Atsuch positions, in terms of deﬁnition (26), the dissipative coupling constant of MSI can beevaluated as follows (see Appendix D) | g γ | = r ms ω c l (cid:112) T ms (34)where l is the eﬀective optical length of the cavity, r ms is the modulus of the amplitudereﬂection coeﬃcient of the membrane, and T ms is the power transmission coeﬃcient of theeﬀective mirror. This result can be compared with the dissipative coupling constant of MOSat | Φ | = Φ , which, via (27), reads | g γ | = 2 ω c l t t m . (35)Clearly for MSI, | g γ | is always appreciably smaller than ω c l while, for MOS, | g γ | can beappreciably larger than ω c l .However, for a balanced comparison, it is reasonable to use the optomechanical coopera-tivity, which can serve as a ﬁgure of merit for optical squeezing and position measurements.Consider the device pumped with a strong monochromatic light ( ω L is its frequency and a is the number-of-photons-normalized amplitude of the pumping ﬁeld inside the cavity).For MSI as a one-sided cavity in the dissipative coupling regime , such a cooperativity, via(34), reads C = ( g γ x zpf a ) γ ms γ m (cid:18) ωγ ms (cid:19) = 2 M r (cid:18) ωγ ms (cid:19) , (36) γ ms = cT ms l , (37)where M comes from (33). Here ω = ck − ω L where ck is the frequency of the detected light.Typically, ω is close to the mechanical resonance frequency ω m . Relation (36) is written forthe bad cavity regime, i.e. for ω m (cid:28) γ ms .Equation (36) is to be compared with the result for MOS given by (32) where, since thedissipative coupling associated with the ”non-feeding” mirror, the small sideband resolvefactor (cid:16) ωγ ms (cid:17) is absent . Comparing (32) with (36), a potential advantage of MOS isclearly seen. 13 IG. 6. Membrane-at-the-edge system: a one-sided cavity with a membrane set inside it, close tothe input mirror. Dashed line - membrane, thin solid line - semitransparent input mirror, thicksolid line - perfectly reﬂecting back mirror, thick arrow - pumping light, thin arrow - detected light.

V. COMPARISON WITH MEMBRANE-AT-THE-EDGE SYSTEM

The membrane-at-the-edge (MATE) system is a one-sided cavity with a mechanical mem-brane placed inside it close to the input mirror as shown in Fig.6. In Ref. 19, variousoptomechanical features of MATE are addressed to demonstrate an advanced optomechani-cal performance in the case of a highly reﬂecting membrane. Speciﬁcally, such an advancedperformance is identiﬁed in a situation where t m (cid:28) λ (cid:28) l and x (cid:28) l t m | g γ | /γ is maximal are identiﬁed, with | g γ | reachingthe value given by Eq.(35).This conclusion also readily follows from our results from Sec. II. Indeed, MATE can beviewed as an optomechanical cavity containing the same synthetic mirror as in MOS, which,however, faces the inner part of the cavity with the opposite side. The power transmissionof such a synthetic mirror is the same in both directions (see Appendix A) such that, undercondition (38), all results obtained in Sec. II for the decay rate and dissipative coupling14onsonant of MOS hold for MATE system (see Appendix E). Next, combining (27) and(25) we ﬁnd that | g γ | /γ reaches maxima at Φ = Φ , leading to the value of | g γ | given byEq. (35).At the same time, there is no reason to expect that the dispersive coupling constant ofMATE will vanish at Φ = Φ , since the amplitude reﬂection coeﬃcients of the syntheticmirror are not the same for the opposite directions (see Appendix A) and, in addition, inthe case of MATE, the length of the inner part of the cavity is not ﬁxed. Moreover, as shownin Appendix E, at Φ = Φ , the system is dominated by the dispersive coupling.Being interested in the situation where the dispersive coupling is absent, one can show(see Appendix E) that it occurs at Φ = Φ , implying, via Eq. (27), | g γ | = ω c l t t m , (39)for t m (cid:28)

1. Thus, comparing this result with (35), one concludes that, in terms of thedissipative coupling constant, MATE is less advantageous than MOS.The same conclusion holds in terms of the cooperativity. Indeed, at Φ = Φ , Eq. (25)yields γ mate = ct l , (40)for the MATE decay rate, implying C = ( g γ x zpf a ) γ mate γ m (cid:18) ωγ mate (cid:19) = M t t m (cid:18) ωγ mate (cid:19) (41)for the optomechanical cooperativity in the bad cavity regime. A comparison of this resultwith (32) is clearly in favour of MOS. VI. CONCLUSIONS

We have theoretically addressed an optomechanical system which consists of a two-sidedcavity and a membrane that is placed outside of it, close to one of its mirrors, while the cavityis fed from the other mirror, and the light leaving it through this mirror being detected.We term such a setup as membrane-outside system (MOS). We have shown that, if themembrane is less reﬂecting than the adjacent mirror and it is positioned very close thepoint ˜ x where the transparency of the mirror/membrane tandem is maximal, the dispersive15oupling can be fully suppressed while the dissipative coupling constant can be potentiallyrecord high. Speciﬁcally, if t m < t (cid:28) t m (cid:28) , (42)where t and t m are the absolute values of amplitude transmission coeﬃcients of the membraneand the mirror, respectively, and the membrane is displaced from ˜ x by δx = λ t m π (43)the system is governed by the dissipative optomechanical interaction with the couplingconstant, which exceeds the dispersive coupling constant for an optomechanical cavity ofthe same length.MOS enables an eﬃcient realization of the two-port conﬁguration, which was recentlyproposed as a promising optomechanical system, allowing among other beneﬁts, e.g., apossibility of quantum limited optomechanical measurements in a system, which does notsuﬀer form any optomechanical instability. Such a setup also enables a kind of switchingbetween the regimes where the quantum limited optomechanical measurements are possibleand where they are not. It is shown that manifestation of that switching is robust to thepresence of an appreciable intracavity loss.The optomechanical performance of MOS is compared with that of other systems, wherethe dissipative coupling is viewed as strong: with the Michelson-Sagnac interferometer(MSI) and with the so-called ”membrane-at-the-edge” system (MATE) . This com-parison is performed in terms of the dissipative coupling constant and optomechanical co-operativity for the regime where the dispersive coupling is absent. It is found that, for anoptimised set of these parameters, the optomechanical performance of MOS is advantageousin both aspects.All in all we have identiﬁed a system, which, among all the systems dominated by thedissipative optomechanical coupling, exhibits the strongest optomechanical interaction. VII. ACKNOWLEDGEMENTS

ESP acknowledges the support of Villum Investigator grant no. 25880, from the VillumFoundation and the ERC Advanced grant QUANTUM-N, project 787520.16

IG. 7. A synthetic mirror consisting of a semitransparent mirror (solid line) and a semitransparentmembrane (dashed line). Running electromagnetic waves are schematically shown with arrows andlabeled with their complex amplitudes.

Appendix A: The scattering matrix of the synthetic mirror

The synthetic mirror in question is schematically depicted in Fig.7. It consists of asemitransparent mirror shown with the solid line and a semitransparent membrane shownwith the dashed line. Their scattering parameters are given by Eqs. (1) and (2). Thecomplex amplitudes of the wave G , G , G , U , U , and U , which are shown in Fig.7, arelinked by the following relations G = itU − rU ,G = − rU + itU ,U = t m e iϕ t G + r m e iϕ r − ikx G ,U e − ikx = r m e iϕ r +2 ikx G + t m e iϕ t G , (A1)where all amplitudes are taken at the mirror. The wave vector of the light is denoted as k .17e are looking for the scattering matrix M of the whole system, which is deﬁned asfollows U G = M U G . (A2)Equations (A1) readily imply M = 11 + rr m e iψ itt m e iϕ t e iϕ r ( r + r m e − iψ ) − r − r m e iψ itt m e iϕ t , (A3)where ψ = 2 kx + ϕ r . Appendix B: Applicability of the synthetic mirror approach to MOS

Conﬁder a two-sided cavity of a ﬁxed length l , where one of the mirrors is reﬂectingwith the π phase shift while the other one is synthetic. We would like to ﬁnd out how thinthe tandem mirror/membrane should be to justify the applicability of the above syntheticmirror approach.

1. Dispersive coupling constant

One readily checks that the resonance frequencies of the system ω c are equal to ck c , where k c satisﬁes the following equation2 lk = π + 2 πN − µ ( kx ) . (B1)Here N is integer and µ ( kx ) is the phase shift at the synthetic mirror. In view of Eqs. (5)and (6), µ a function of kx . Equation (B1) implies dω c dx = − ω c µ (cid:48) l

11 + xµ (cid:48) l , (B2) µ (cid:48) = dµd ( kx ) . Taking into account that, according to Eq. (23) in the range of interest, | µ (cid:48) | is about 8 t /t m or smaller, we conclude that, if x (cid:28) l t m t , (B3)18he second fraction in (B2) can be replaced with 1 to yield dω c dx = − c l dµdx . (B4)Equations (B3) and (B4) bring us to Eqs. (29) and (28) from the main text.

2. Decay rate and dispersive coupling constant

In terms of the complex amplitudes (see Fig. 7), the decay rate associated with thesynthetic mirror can be written as follows γ = ˙ WW = ct m | G | l | U | + x | G | ) = ct m l | U | / | G | + x ) , (B5)where W is the energy stored in the system and ˙ W is the dissipated power. Equations (A1)and (4) imply | U | | G | = 1 + r r m + 2 rr m cos ψt = t m T (B6)such that Eq. (B5) can be rewritten as follows γ = cT l − xl Tt m . (B7)Taking into account that, in the case of interest, T is about 4 t /t m or smaller(see Eq. (21)),we conclude that, if x is small enough such that inequality (B3) is satisﬁed, the secondfraction in (B7) can be replaced with 1 to justify Eq.(14) from the main text.Next, Eq. (B7) yields dγdx = ct m lt m T dTdx lt m /T + x ) . (B8)Using (21) and (22), in the case of interest, lt m T dTdx can be evaluated as 4 kl/T (cid:29) x is smallenough such that inequality (B3) is satisﬁed, Eq. (B8) can be rewritten as follows dγdx = c l dTdx ,justifying the calculation of the dissipative coupling constant by using the synthetic mirrorapproach. Appendix C: The backaction-imperfection product in the presence of intracavitylosses

To evaluate the impact of the intracavity losses on the backaction-imperfection product ofa two-port cavity, we model the intracavity losses as the third port. The system is pumped19ith a strong coherent light of frequency ω L from the ﬁrst port, the light backscatteredfrom this port is detected. We describe the ﬂuctuations in the system with the followingequations written for the Fourier transforms of all variables (the argument ω is dropped) inthe reference rotating with frequency ω L : (cid:20) γ + γ + γ − iω (cid:21) X + ∆ Y = √ γ X in1 + √ γ X in2 + √ γ X in3 + a g γ x , (C1) (cid:20) γ + γ + γ − iω (cid:21) Y − ∆ X = √ γ Y in1 + √ γ Y in2 + √ γ X in3 + a g ω x , (C2) F = − a (cid:126) g γ √ γ Y in2 + 2 a (cid:126) g ω X . (C3)where g γ and g ω are deﬁned by (26). Here ∆ = ω L − ω c , where ω c is the resonance frequency, γ , , are the decay rates of the three ports, a is a number-of-photon normalized amplitudeof the intracavity pumping ﬁeld. The operator of mechanical displacement is denoted as x .The quadratures of operators of ﬂuctuating parts of the intracavity ﬁeld, a , and those of theinput ﬁelds, A in1,2,3 , are deﬁned as follows X ( ω ) = [ a ( ω ) + a † ( − ω )] / , Y ( ω ) = − i [ a ( ω ) − a † ( − ω )] / , X in1,2,3 ( ω ) = A in1,2,3 ( ω ) + A † in1,2,3 ( − ω ) , Y in1,2,3 ( ω ) = − i [ A in1,2,3 ( ω ) − A † in1,2,3 ( − ω )] . The correlators of the ﬁeld quadratures satisfy the following relations (cid:104) X in1,2,3 ( ω ) X in1,2,3 ( ω (cid:48) ) (cid:105) = (cid:104) Y in1,2,3 ( ω ) Y in1,2,3 ( ω (cid:48) ) (cid:105) = i (cid:104) Y in1,2,3 ( ω ) X in1,2,3 ( ω (cid:48) ) (cid:105) = − i (cid:104) X in1,2,3 ( ω ) Y in1,2,3 ( ω (cid:48) ) (cid:105) = δ ( ω + ω (cid:48) ) , (C4)where (cid:104) ... (cid:105) stands for the ensemble averaging.The output ﬁeld from the ﬁrst port, which is detected, obeys the following relation X in1 + X out1 = 2 √ γ X , Y in1 + Y out1 = 2 √ γ Y . (C5)We are interested in the backaction-imperfection product for the symmetric two-sidedcavity ( γ = γ = γ ), the resonance excitation (∆ = 0), and in the low frequency limit( ω/γ ⇒ X out1 = ˜ X in + 2 a g γ √ γγ + γ / x , (C6)20 out1 = ˜ Y in + 2 a g ω √ γγ + γ / x , (C7)where the input noise operators ˜ X in and ˜ Y in evidently meet Eqs. (C4).The optimal quantum-mechanical measurements must employ the quadrature Z out = X out1 cos θ + Y out1 sin θ such that the orthogonal quadrature carries no information about x . This condition is met at θ = tan − ( g ω /g γ ). For the optimal quadrature, we ﬁnd Z out = Z in + a √ γ (cid:113) g γ + g ω γ + γ / x , (C8)where the input noise operator Z in obeys relations (C4), implying the following spectralpower density for the imprecision of position measurements S imp xx = ( γ + γ / a γ g γ + g ω . (C9)In the situation considered, for the stochastic backaction force, Eqs. (C1) and (C3) yield F = − (cid:126) a √ γ g γ Y in2 + (cid:126) a g ω √ γγ + γ / X in1 + X in2 + X in3 (cid:112) γ /γ ) , (C10)which, via (C4), leads to the following expression for the spectral density of this force S F F = (cid:126) a γ ( γ + γ / (cid:34)(cid:18) γ γ (cid:19) g γ + 2 (cid:18) γ γ (cid:19) g ω (cid:35) . (C11)Combining (C9) and (C11), we arrive at the following backaction-imperfection product S imp xx S F F = (cid:126) A + 2 Aξ ξ , ξ = g ω g γ , A = 1 + γ γ , (C12)which is given by Eq.(31) of the main text. Appendix D: Michelson-Sagnac interferometer

The Michelson-Sagnac interferometer (MSI) is schematically depicted in Fig. 5. It consistsof a beam splitter, a membrane and three perfectly reﬂecting mirrors. The beam splitterand the membrane are characterized by following scatting matrices T b − R b R b T b and − r ms t ms t ms r ms , (D1)21espectively, where all coeﬃcients of the matrices are real and positive; t ms and T b standfor the amplitude transmission coeﬃcients. The membrane is displaced to the left fromits symmetric position by the distance x . According to Ref. 10, MSI can be treated as anoptomechanical cavity of a ﬁxed length l with the input mirror, the scattering matrix ofwhich reads M = ρ ττ − ρ ∗ , ρ = | ρ | e iµ , (D2) ρ = − R b T b t ms − ( R b − T b ) r ms cos 2 kx + ir ms sin 2 kx, (D3) τ = t ms ( T b − R b ) + 2 R b T b r ms cos 2 kx, (D4)where τ stands for the amplitude transmission coeﬃcient, while, for the decay rate and theoptomechanical coupling constants, the following relations can be used: γ ms = cT ms l , T ms = τ (D5)for the decay rate and g ω = − dω c dx = dµdx c l , (D6) g γ = − dγ ms dx = − τ dτdx c l (D7)for the coupling constants, where dτdx = − kr ms R b T b sin 2 kx, dµdx = − kr ms [2 t ms R b T b cos 2 kx − r ms ( T b − R b )] . (D8)We are interested in the values of g γ and γ ms for the position x of the membrane wherethe dispersive coupling vanishes. According to (D6), this happens when dµdx = 0, implyingvia Eq. (D8) the condition for x , which readscos 2 kx = r ms T b − R b t ms R b T b . (D9)Under this condition, according to Eq. (D4) τ = T b − R b t ms . (D10)For the validity of our calculations, we need | τ | (cid:28)

1, yielding T b ≈ R b ≈ | cos 2 kx | = | r ms τ | (cid:28) . (D11)To be speciﬁc, we will work close to the point where 2 kx ≈ π/

2. Then Eq. (D8) implies ∂τ∂x ≈ − kr ms (D12)and (cid:12)(cid:12)(cid:12)(cid:12) dγ ms dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) dτ dx (cid:12)(cid:12)(cid:12)(cid:12) c l ≈ | T b − R b | r ms t ms ω c l = 2 (cid:112) T ms r ms ω c l . (D13)Equations (D7) and (D13) bring us to Eq. (34) of the main text. Appendix E: Membrane-at-the-edge system1. Vanishing of the dispersive coupling

For MATE, we are interested in the position of the membrane where the dispersivecoupling vanishes. Solving the following well-known resonance equation cos( kl + ϕ r ) = − r m cos(2 kx − kl ) , (E1)we ﬁnd 2 x − l = 1 k (cid:20) ± cos − (cid:18) cos( kl + ϕ r ) r m (cid:19) + 2 πN (cid:21) , (E2)where N is an integer, and calculate dk/dx at the resonance values of k , k = ω c /c : (cid:18) dkdx (cid:19) − = l k (cid:34) − xl ± r − m (cid:115) t m cos ( kl − kx )1 − cos ( kl − kx ) (cid:35) . (E3)Equation (E3) implies that the dispersive coupling vanishes, i.e. dω c /dx = 0, at the reso-nance wave vector satisfying the following conditioncos ( kl − kx ) = 1 , (E4)or, alternatively, after some algebra, atcos(2 kx + ϕ r ) = − r m . (E5)In the case of interest where r m is close to 1, Eq. (E5) implies that Φ deﬁned by Eq.(20) issmall such that Eq. (E5) yields − / − t m / . (E6)23he solution to this equation reads Φ = t m / , (E7)which is the result used in the main text.

2. Condition on x for the enhanced optomechanical performance of MATE Let us ﬁnd the condition on x , enabling the enhanced value of dispersive coupling constantof MATE identiﬁed in Ref. 19. For λ (cid:28) l and x (cid:28) l , according to Eq. (E3), the maximummodulus of the dispersive coupling constant is reached at cos( kl − kx ) = 0 while taking” − ” in this formula. Such a maximum value reads g ω = dω c dx = ω c x + lt m / . (E8)This relation implies that the aformentioned enhanced value of the dispersive coupling con-stant of MATE, which is equal to 4 ω c / ( lt m ), corresponds to x (cid:28) l t m

3. Dispersive coupling at

Φ = Φ According to Eq. (E3), to evaluate the dispersive coupling constant at Φ = Φ , it suﬃcesto know cos ( kl − kx ). To ﬁnd it, we note that Eq. (E1) can be rewritten as followstan( kl − kx ) = r m + cos(2 kx + ϕ r )sin(2 kx + ϕ r ) , (E10)while, at Φ = Φ , and t (cid:28) t m (cid:28)

1, Eq. (9) impliescos(2 kx + ϕ r ) = − r m r m , sin(2 kx + ϕ r ) = ± − r m r m . (E11)Combining the above relations we ﬁndcos( kl − kx ) = 11 + tan( kl − kx ) = 11 + r m = 1 / , (E12)24eading, for the two modes corresponding to ± in (E3), to the following expressions for thedispersive coupling constants g ω = − dω c dx = − ω c l − x + lt m / g ω − = dω c dx = ω c x + lt m / , (E14)respectively.The mode exhibiting coupling constant given by Eq. (E14) is relevant to our consideration.The reason is as follows. The spectrum of the whole cavity in the k − x plane is, actually,made of the resonance curves of its two parts with small areas of the avoided crossing.Evidently, the dispersive coupling constant of the resonance curves originating from theresonance curves for the x -long part is positive while the dispersive coupling constant of theresonance curves originating from the resonance curves for the l − x -long part is negative.Addressing Φ (cid:28)

1, we are close to the line given by equation cos(2 kx + ϕ r ) = −

1, which isthe resonance curve for the x -long part. Thus, we conclude that, for Φ (cid:28)

1, the dispersivecoupling constant should be positive as that given by Eq. (E14) is.Next, in view of condition (E9), Eq. (E14) yields g ω − = dω c dx = ω c l t m (E15)and, ﬁnally, combining (35) and (E15) we ﬁnd (cid:12)(cid:12)(cid:12)(cid:12) g γ g ω − (cid:12)(cid:12)(cid:12)(cid:12) = t t m = T (cid:28) , (E16)where Eq.(21) is taken into account. Equation (E16) implies that, at Φ = Φ , MATE isdominated by the dispersive coupling.

4. Applicability of the synthetic mirror approach to MATE

Let us show that under conditions (E9) and λ (cid:28) l , the results for the decay rate anddissipative coupling constant obtained in Sec.II using the synthetic mirror approach can beapplied to MATE.According to Ref. 19, for t (cid:28) t m , the decay rate of MATE reads γ mate = ct t m / xt m + ( l − x )[1 + r m + 2 r m cos(2 kx + ϕ r )] , (E17)25hich can be rewritten as follows γ mate = cT l

11 +

A , A = xt m r m + 2 r m cos(2 kx + ϕ r ) , (E18)where T comes from Eq. (4). In the situation of interest, where cos(2 kx + ϕ r ) ≈ − A = 4 x/ ( lt m ) (cid:28) t (cid:28) t m , the dissipative coupling constant of MATE reads dγ mate dx = ct t m r m + r m cos(2 kx + ϕ r ) + 2 r m k ( L − x ) sin(2 kx + ϕ r ) { l [1 + r m + 2 r m cos(2 kx + ϕ r )] − x [ r m + r m cos(2 kx + ϕ r )] } , (E19)Here, as was shown just above, condition (E9) enables dropping of the second term in thedenominator while, for cos(2 kx + ϕ r ) ≈ − − t m / kl sin(2 kx + ϕ r ) . In the present text, we discuss MATE for Φ ≥ Φ implying, via (E11), | sin(2 kx + ϕ r ) | ≥ t m / λ (cid:28) l . Thus weﬁnd dγ mate dx = ckl t t m sin(2 kx + ϕ r )[1 + r m + 2 r m cos(2 kx + ϕ r )] . (E20)This relation is consistent with the result given by Eq. (15) and (16), which is obtainedusing the syntectic mirror approach. ∗ [email protected]ﬂ.ch M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. , 1391 (2014). F. Elste, S. M. Girvin, and A. A. Clerk, Phys. Rev. Lett. , 207209 (2009). S. Huang and G. Agarwal, Physical Review A , 023844 (2017). T. Weiss, C. Bruder, and A. Nunnenkamp, New Journal of Physics , 045017 (2013). T. Weiss and A. Nunnenkamp, Phys. Rev. A , 023850 (2013). D. Kilda and A. Nunnenkamp, Journal of Optics , 014007 (2016). S. P. Vyatchanin and A. B. Matsko, Physical Review A , 063817 (2016). A. Nazmiev and S. P. Vyatchanin, Journal of Physics B: Atomic, Molecular and Optical Physics , 155401 (2019). N. Vostrosablin and S. P. Vyatchanin, Phys. Rev. D , 062005 (2014). S. P. Tarabrin, H. Kaufer, F. Y. Khalili, R. Schnabel, and K. Hammerer, Phys. Rev. A ,023809 (2013). A. Xuereb, R. Schnabel, and K. Hammerer, Phys. Rev. Lett. , 213604 (2011). A. K. Tagantsev, I. V. Sokolov, and E. S. Polzik, Phys. Rev. A , 063820 (2018). A. K. Tagantsev and S. A. Fedorov, Physical review letters , 043602 (2019). A. Mehmood, S. Qamar, and S. Qamar, Physica Scripta , 095502 (2019). F. Y. Khalili, S. P. Tarabrin, K. Hammerer, and R. Schnabel, Phys. Rev. A , 013844 (2016). S. Huang and A. Chen, Physical Review A , 063818 (2018). G. Huang, W. Deng, H. Tan, and G. Cheng, Physical Review A , 043819 (2019). A. Mehmood, S. Qamar, and S. Qamar, Physical Review A , 053841 (2018). V. Dumont, S. Bernard, C. Reinhardt, A. Kato, M. Ruf, and J. C. Sankey, Optics express ,25731 (2019). A. K. Tagantsev, Physical Review A , 063813 (2020). A. K. Tagantsev, Physical Review A , 043520 (2020). M. Li, W. H. P. Pernice, and H. X. Tang, Phys. Rev. Lett. , 223901 (2009). A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K. Hammerer, and R. Schn-abel, Phys. Rev. Lett. , 043601 (2015). V. Tsvirkun, A. Surrente, F. Raineri, G. Beaudoin, R. Raj, I. Sagnes, I. Robert-Philip, andR. Braive, Scientiﬁc reports , 16526 (2015). M. Wu, A. C. Hryciw, C. Healey, D. P. Lake, H. Jayakumar, M. R. Freeman, J. P. Davis, andP. E. Barclay, Phys. Rev. X , 021052 (2014). H. M. Meyer, M. Breyer, and M. K¨ohl, Applied Physics B , 290 (2016). M. Zhang, A. Barnard, P. L. McEuen, and M. Lipson, in

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