Two-membrane cavity optomechanics

We study the optomechanical behaviour of a driven Fabry-P\'erot cavity containing two vibrating dielectric membranes. We characterize the cavity-mode frequency shift as a function of the two-membrane positions, and report a $\sim 2.47$ gain in the optomechanical coupling strength of the membrane relative motion with respect to the single membrane case. This is achieved when the two membranes are properly positioned to form an inner cavity which is resonant with the driving field. We also show that this two-membrane system has the capability to tune the single-photon optomechanical coupling on demand, and represents a promising platform for implementing cavity optomechanics with distinct oscillators. Such a configuration has the potential to enable cavity optomechanics in the strong single-photon coupling regime, and to study synchronization in optically linked mechanical resonators.


Introduction
Multi-element systems of micro/nano-mechanical resonators offer promising prospects for enhanced optomechanical performances [1][2][3][4][5][6][7], coherent control [8,9], and for the exploration of multi-oscillators synchronization [8,[10][11][12][13][14][15][16]. The standard path for reaching the strong single-photon optomechanical coupling regime is to consider co-localized optical and vibrational modes [17][18][19], with a large spatial overlap confined in very small volumes, corresponding to mechanical modes with extremely small effective mass. An alternative solution, capable of providing systems with orders of magnitude increased ratio between the single-photon optomechanical coupling rate, and the cavity decay rate, is to exploit quantum interference in multi-element optomechanical setups [3][4][5]. Although the simplest two-membrane sandwich in an optical cavity is a paradigm for the realization of strong-coupling optomechanics, and the observation of collective mechanical effects (such as synchronization), no experimental studies of these phenomena have been reported till now. Previous related results [20] were confined only to the optical and mechanical characterization of two-membrane sandwiches.
Here we report on the first experimental characterization of the optical, mechanical, and especially optomechanical properties of a sandwich constituted of two parallel membranes within an optical cavity. We show how the resonance frequencies of the optical cavity are shifted as a function of the position of the two membranes. This effect is central to the description of the optomechanical properties of the system, since it provides a direct estimation of the strength of the couplings [1,[21][22][23]. By investigating the shifts of the cavity resonances we find that the optomechanical coupling strength is enhanced by constructive interference when the two membranes are positioned to form an inner cavity which is resonant with the driving field. Specifically we determine a gain of ∼2.47 in the coupling strength of the relative mechanical motion with respect to the single membrane configuration. We finally prove both the capability to tune on demand the single-photon optomechanical couplings, and the simultaneous optical cooling of the fundamental modes of the two distinct membranes.

Theory
Generalizing the results obtained in [5], we consider the case of two different movable dielectric membranes placed inside a Fabry-Pérot cavity of length L, which is driven by an external laser. The Fabry-Pérot cavity is composed of two identical mirrors with electric field reflection and transmission coefficients r and t, respectively. The membranes can be modelled as dielectric slabs of thickness L j m, and index of refraction n j (where the index j=1, 2 distinguish the parameters of the two membranes), such that their reflection and transmission coefficient can be expressed as where k=2π/λ is the wavenumber of the electric field, and λ is its wavelength. The optical resonance frequencies correspond to the maxima of transmission of the whole cavity. The electric field amplitudes A j of incident ( j=in), reflected ( j=ref), and transmitted ( j=tran) waves, as well as for the fields in the cavity ( j=1, 2, K, 6) (see figure 1), satisfy the following equations: where L j =q j −q j−1 (with q j the positions of the various elements defined in figure 1, and j=1, 2, 3) is the length of the subcavities formed by the mirrors and the membranes, so that L=L 1 +L 2 +L 3 . We point the reader to [22] for a similar approach in the case of a single membrane. Here we use the same convention of [22] for the scattering matrix of a single scattering element, either the cavity mirror or the membrane. This is a bit different from the choice of [5], which is reproduced by replacing r with −r into the equations above. Equations (1)-(10) are valid, for any value of the thickness, in the ideal one-dimensional case of plane waves, and flat, aligned mirrors and membranes. They can be applied also to the case of Gaussian cavity modes and spherical external mirrors as long as the membranes are placed within the Rayleigh range of the cavity. The system of equations (3)-(10) can be solved to determine the transmission coefficient of the whole cavity. It is given by  Figure 1. Schematic diagram of the system. Two movable dielectric membranes are placed inside a Fabry-Pérot cavity of length L which is driven by an external laser. The position of two fixed mirrors (movable membranes) is denoted by q 0 and q 3 (q 1 and q 2 ); we  2  2  2 2i  2 1  2  1  2 2i   2  1 2  2i  1 2  2  2  2 2i   1  2i  1 2  2i  2  2i   1  2   1  3  2  3   1  2  3 This last expression reproduces equation (4) of [5] when r 1 =r 2 , t 1 =t 2 , and we restrict to the case of real n j , implying in particular 2i j . Moreover it reproduces also the case of a single membrane which is obtained by taking . From equations (3)-(10) one can also derive the expression for the reflectivity, given by The cavity mode frequencies correspond in general to the maxima of the transmission, and therefore the minima of  | | 2 . In the limiting case of perfect external mirrors, R=1, these maxima become poles of the transmission and the modes correspond to the zeros of . In order to get a simple expression for the poles we restrict to this limiting situation which, as we have seen in [5], works also in the case of realistic high-finesse cavities for which typically 1−R∼10 −5 . In particular using the definitions = + L q L 2 By setting this equation equal to zero we get the implicit equation for the cavity mode frequencies valid in the limit of R∼1 and for the general case of two different membranes. It reproduces the implicit equation in the two special cases of equal membranes and of one membrane only. Specifically, in the case of equal membranes R 1 =R 2 =R m , f 1 =f 2 =f, and using the definitions , where Q=(q 1 +q 2 )/2 is the center-of-mass (CoM) coordinate, we get and which is just the corresponding equation used in [24] in the limit R=1.
In the general case the implicit equations for the mode frequencies  = 0, with  given in equation (15), can be expressed using the definitions . This can be further simplified with the definitions such that equation (17) can be rewritten in the equivalent form where and which, in turn, is equivalent to its formal solution obtained by inverting the sin function, that, using the definition of ¢ L , can be expressed as 1 a r c s i n , and ℓinteger. For each value of ℓone finds a solution for a cavity mode wavenumber k ℓ that can be decomposed as the sum, 0 ) and the shift due to the membranes that is given by the implicit expression , so that ℓis a very large integer and this implies d  In this limit one can safely take Correspondingly, for R 1 and R 2 not too close to one, and for not too large values of q 1 and q 2 , i.e., when q 1 /L, q 2 /L=1, (see [5]), one can safely express the shift explicitly as a function of the empty cavity solution as , that can be also written as an equation for the cavity mode frequency This treatment in the general case of two different membranes generalizes previous results and has the advantage of providing a unique framework in which one can immediately compare the single and two-membrane case.
On the other hand, for a given value of the maximum available membrane reflectivity R max =max{R 1 , R 2 }, we have numerically verified that the largest optomechanical couplings are achieved when the two membranes have identical reflectivities. For this reason we have focused our experiments to the case of nominally identical membranes, and we shall restrict from now on to this latter case. In particular, introducing the parameters m , 2 and n=n 1 =n 2 , the explicit dependence upon the variables kq 1 and kq 2 of the parameters  ( ) kq kq , 1 2 and q ( ) kq kq , 1 2 that enter into the definition of  in equation (22), is easily obtained from equations (19) and (20), so that for identical membranes one has The optomechanical couplings strength G j are the derivative of the optical mode frequencies with respect to the position of the jth membrane q j . Defining the scaled dimensionless positions In the case of a single membrane the single-photon optomechanical coupling has the same structure of equation (26)  l sing sing but with a different dimensionless frequency shift function Taking the derivative one can see that the maximum value of  p ¶ ¶ (˜)q q 2 sing is  4 m (halfway between a node and an antinode of the field), so that In order to study the enhancement of the coupling (and the associated optical interference effect) due to the presence of the second membrane, we have to compare the maximum derivative of the function  p p (˜˜) q q 2 , 2 1 2 with respect to  4 m . In figure 2 we show the cavity mode frequency shifts, and superimposed the vector plot of the corresponding gradient field, which gives the values of the two couplings G 1 and G 2 . It shows that the largest optomechanical coupling is achieved simultaneously by the two membranes, and in this case G 1 =−G 2 . At this point the cavity mode frequency is sensitive at first order only to the variation of the distance between the two membranes, q=q 2 −q 1 , and is not sensitive to shifts of the CoM of the two membranes, Q. This implies that the coupling of the CoM is zero, G Q =0, while that of the relative coordinate is = | | | | G G q j [5]. In this case, in order to determine the gain factor we apply the same argument of section III of [5] from equations (19)- (23). Specifically, we find that, for ℓ integer This means that the maximum coupling for both membranes is achieved when + (˜˜) q q 1 2 is an integer number for even ℓ, and an half-integer for odd ℓ (and this is visible also from the vector plots in figure 2). Using this condition equation (30) reduces to In the case of  = 0.4 m , as in our experiment, the optomechanical coupling may increase up to a factor ∼2.72. As discussed in detail in [5] (see also [3,4]), the present treatment based on the assumption d  ℓ ℓ ( ) k k 0 , allowing to express the frequency shift explicitly as a function of the empty cavity solution (see equation (23)), is valid provided that the reflectivity  m is not too close to one. This fact could be guessed from the fact that equations (30)- (31) suggest an unlimited value of the optomechanical coupling when   1 m , which is unphysical. In fact, as numerically shown in [5] and could be expected also on physical grounds, when  R 1 m (that is, the membrane reflectivity becomes equal or larger than the cavity mirror reflectivity), equation (30) is no more valid, and the optomechanical coupling saturates to a value corresponding to that of the inner Fabry-Perot membrane cavity with length . As underlined in [5], when  | | q L 1 and ~R 1 m , this saturation value would still correspond to the strong-coupling regime where the single-photon optomechanical coupling is equal or larger than the cavity decay rate, because for aligned membranes with negligible absorption, the cavity decay rate remains identical to the value of the main cavity

Membrane-sandwich characterization
In our experiment we used two different membrane sandwiches. The first is constituted of two low-stress SiN square membranes, with a side of 1 mm, and a thickness of 100 nm. And the second is made of two high-stress Si N 3 4 square membranes, with a side of 1.5 mm, and a nominal thickness of 100 nm. In both cases, one of the membranes is glued on a piezo, which allows for a scan of the membrane-cavity length, while the whole membrane-cavity mount is attached to another piezo in order to displace in a controlled way the CoM of the two membranes.

Optical properties
Here we report on the characterization of the two-membrane sandwiches in terms of reflectivity  m and cavity length L c , which we have performed before inserting them into the optical cavity. In particular, the membranecavity length L c was determined by illuminating the membrane-sandwich with a tungsten lamp. The transmitted light was collected by a multi-mode fiber, and finally revealed by a spectrometer. The interference pattern of the normalised transmitted light is shown in figures 3(b) and (c), for the first and second sandwich, respectively, and compared with a best-fit curve obtained from the expression of the transmitted light where  in is the input light intensity, Δ=4π L c /λ, and  is the finesse of the membrane-cavity. From the spectrometer data of the first sandwich, figure 3(b), we obtain a best-fit value for the membrane-cavity length m =  L 24.008 0.004 m c . Moreover, assuming a finesse given by the equation , which is found for the index of refraction of Si 3 N 4 given in [25]. Although the membrane-cavity length is well estimated by the peak distances in the interference patterns reported in figures 3(b) and (c), the membrane thickness, and consequently the reflectivity of the membrane, is badly derived by the poor visibility of the curves, measured with an apparatus not optimized for this purpose. The membrane reflectivity  m at specific wavelengths is optimally estimated with a different experiment (see figure 4(a)) exploiting again equation (32) and (33), but now collecting on a photodiode the light of a laser transmitted through the membranecavity while scanning the cavity length q=L c +δq, such that, in this case, we use Δ=4π q/λ in equation (32). For the first sandwich we use a 1064 nm laser, and the best-fit provides a value of the finesse  =  This result is estimated by using the values of the refractive index at the three wavelengths reported in [25], which are in accordance with the ones provided by the manufacturer.

Mechanical properties
Here we present a study of the mechanical properties of the second membrane-sandwich by using a 532 nm laser in a Michelson interferometer, as shown in figure 5 [26] (this kind of study is not possible with the first sandwich due to the poor quality of the mechanical modes). In figure 6 we show the thermal voltage noise (VSN) of the two-membranes cavity revealed by homodyne detection of the reflected light, the quality factor m of the mechanical modes, and the relative difference between experimental and fitted mechanical frequencies. The membranes are very similar and show a set of very close resonance peaks. As shown in figure 6(b), we reproduced the mechanical resonance frequencies of both membranes with an error smaller than 1% assuming rectangular membranes and the nominal values provided by the manufacturer for the stress, s = 0.825 GPa, and for the density r = . Figure 6(c) shows that the mechanical quality factor changes significantly between the modes and that one membrane tends to have lower  m values. We attribute these scattered values to the effect of clamping which strongly depends upon the shape of the vibrational mode and may be different on the two membranes with the current mounting.

Estimation of the optomechanical coupling strength
In order to estimate the strength of the optomechanical coupling achievable with our system we have inserted the first sandwich (the one made with the SiN membranes) in a 90 mm-length optical cavity [27,28], and the optomechanical system was located in a vacuum chamber evacuated to´-5 10 mbar 7 (see figure 7). Our aim is to compare the frequency shift of the resulting cavity modes in the presence of the two-membrane system, with the one corresponding to the case with a single membrane inside. We note that the results for a single membrane are obtained using a membrane different form the ones of the sandwich, namely a highly stressed SiN circular membrane, with a diameter of 1.2 mm, and a thickness of 97 nm [24,27,28]. However, the fact that the membranes have similar size and are made of the same material, makes the comparison that we report hereafter meaningful.
The spectra of the cavity modes reported in figures 8 and 9 are obtained by detecting the light of a laser at 1064 nm transmitted by the cavity while scanning the laser frequency for different positions of the membrane(s). The last panel on the right of figure 8 is equal to the last of figure 9 and they report the results of the single membrane case. The slope of the corresponding black lines represents the maximum achievable single membrane optomechanical coupling strength p´- G 2 3.47MHznm sing max 1 . The other panels show the results with two membranes. In this case there are two degrees of freedom that can be varied,that is,the positions of the two membranes, q 1 and q 2 . Due to the design of our membrane-cavity, we can scan either the CoM, Q, for different values of the membrane distance =q q q 2 1 ,or q 1 for different positions of q 2 . In figure 8 are reported  figure 9 we report the spectra obtained by scanning the position q 1 for different position q 2 , as indicated by the lines I-VI in   In this case the optomechanical coupling strength increases by a factor~2.47, which is 9% lower than the expected one, given by equation (31). Such a discrepancy might be attributed to an imperfect alignment of the two membranes.

Cavity finesse in the presence of the membrane-sandwich
In the last set of experiments we placed the second membrane sandwich (the one made of Si N 3 4 membranes) in the same optical cavity of figure 7 (see also figure 10(a)). Here we report on the analysis of the effects of the membranes on the cavity finesse. The finesse of the optical cavity, with and without the membrane sandwich, is determined by means of the ring-down technique, fitting the decay of the normalized transmitted intensity,   tr , after the laser at 1064 nm is rapidly turned off. In figure 10(b)  , with pFSR 2 1.67 GHz. The observed reduction of finesse in the presence of the membrane-sandwich is much more significant than the one occurring in the case of a single membrane [24,29] and it can be ascribed to the imperfect alignment of the two membranes [20]. This misalignment is responsible for an effective cavity loss , and a diffraction angle of the gaussian beam 112 m 0 the beam waist of the cavity of our experiment, the misalignment angle q wdg between the two non-parallel membranes can then be estimated to be q m 30 rad wdg . The membrane alignment could be improved either by using pairs of membranes assembled parallel to each other by means of spacers deposited on one of the chip, as implemented for example in the experiment of [20], or by replacing the single Figure 10. (a) Experimental setup for studying cavity optomechanics with a two-membrane setup within a cavity. A laser probe beam, frequency modulated by an electro-optical modulator (EOM), impinges on the optical cavity. The reflected beam is split: one component is detected, demodulated and low-pass amplified for generating the Pound-Drever-Hall (PDH) error signal able to lock the laser to the cavity; the second component is analysed by homodyne detection in order to detect the mechanical motion. A further beam, the cooling beam, detuned by Δ from the cavity resonance, is turned on for engineering the optomechanical interaction, and in particular realize laser cooling of the mechanical modes. HWP denotes a half-waveplate, QWP a quarter-waveplate, BS a beamsplitter, and PBS a polarizing beam-splitter.  , for a cooling input power 83kHz, and g 0 as in figure 11. The red and orange dashed lines indicate the mechanical frequencies with no cooling. (b) Theoretical prediction with parameters given in figure 11.  . Note the less effective optomechanical cooling on the left mode due to lower optomechanical coupling, and also the frequency shift in the moderate resolved-side-band limit. mean mechanical frequency w m , and compare it with the corresponding theoretical prediction (right panels). In figure 12 we use a lower power of the cooling beam with respect to that used in figure 13, but in both cases the agreement is very good. In figure 14 instead we report the DSN as a function of the cooling beam power P C , at a fixed detuning w D~¯m.

Conclusion
We studied the optomechanical behaviour of a driven Fabry-Pérot cavity containing a two-membrane sandwich. From the cavity mode frequency shift as a function of the membrane positions, we derived a ∼2.47 gain in the optomechanical coupling strength with respect to the single membrane case. This is obtained when the two membranes are positioned to form an inner cavity resonant to the driving field. We also showed the capability of the system to be tuned on demand, and the simultaneous optical cooling of the fundamental modes of the two distinct membranes. Such a configuration has the potential to enable cavity optomechanics in the strong single-photon coupling regime [3][4][5], as well as to study the nonlinear dynamics and synchronization of two distinct nano-mechanical resonators by means of an optical link [12][13][14][15][16].