Strong optomechanical interactions in a sliced photonic crystal nanobeam
SStrong optomechanical interactions in a slicedphotonic crystal nanobeam
Rick Leijssen and Ewold Verhagen ∗ Center for Nanophotonics, FOM Institute AMOLF, Science Park 104, 1098 XG,Amsterdam, The Netherlands
AbstractCavity optomechanical systems can be used for sensitive de-tection of mechanical motion and to control mechanical res-onators, down to the quantum level. The strength with whichoptical and mechanical degrees of freedom interact is defined bythe photon-phonon coupling rate g , which is especially largein nanoscale systems. Here, we demonstrate an optomechanicalsystem based on a sliced photonic crystal nanobeam, that com-bines subwavelength optical confinement with a low-mass me-chanical mode. Analyzing the transduction of motion and effectsof radiation pressure we find a coupling rate g / π ≈ . . The broad bandwidth is useful for appli-cation in miniature sensors, and for measurement-based controlof the resonator’s motional state. The motion of a mechanical resonator can be read out with extreme sen-sitivity in a suitably engineered system whose optical response is affected bythe displacement of the resonator. The resultant coupling between optical andmechanical degrees of freedom also gives rise to a radiation pressure force thatenables actuation, tuning, damping, and amplification of the resonator, withapplications ranging from classical information processing to quantum controlof macroscopic objects . Such control can be established either passively, byemploying the intrinsic dynamics of the system , or actively, by using the out-come of displacement measurements . Fast, sensitive measurement of nanome-chanical displacement can as such be used for optical cooling , squeezed lightgeneration , quantum non-demolition measurements and enhancing sensorbandwidth .In a cavity optomechanical system, which has an optical resonance frequency ω c that depends on the position of a resonator, both the sensitivity of a displace-ment measurement and the magnitude of effects caused by radiation pressure ∗ [email protected] a r X i v : . [ phy s i c s . op ti c s ] M a y orces are governed by two parameters: on the one hand the strength with whichacoustic and optical degrees of freedom interact, expressed as the magnitude ofthe resonator’s influence on the frequency ω c , and on the other hand the cavitylinewidth κ . The interaction strength is characterized at the most fundamentallevel by the vacuum optomechanical coupling rate g , as it enters the optome-chanical interaction Hamiltonian ˆ H int = ¯ hg ˆ a † ˆ a (ˆ b + ˆ b † ), where ˆ a and ˆ b are thephoton and phonon annihilation operators, respectively. As this Hamiltonianshows, g is the frequency response of the optical cavity due to the mechanicaldisplacement in a typical quantum state, where the total number of phonons isof the order of 1.Per photon in the cavity, the effective optomechanical measurement rate ,as well as the radiation-pressure induced alteration of a resonator’s frequencyand damping through dynamical backaction, scale with g /κ . Improving thisratio is thus desirable for more sensitive measurements and for better opticalcontrol of the mechanical resonator. Decreasing the optical damping κ to a lowvalue has been very fruitful, but can present several drawbacks as well: narrowlinewidths place stringent demands on excitation sources and fabrication tol-erances, and make integration of many devices, e.g. in practical sensor arrays,difficult. Moreover, dynamical instabilities and nonlinear linewidth broadeninglimit the number of photons with which a high-Q cavity can be populated. Fi-nally, several schemes for measurement and control in fact rely on fast, broad-band optical response . The photon-phonon coupling rate g , vice versa,is given by g = Gx zpf , where G = ∂ω c /∂x is the frequency shift per unitdisplacement x and x zpf = (cid:112) ¯ h/ m eff Ω m are the zero-point fluctuations of aresonator with mass m eff and frequency Ω m . The magnitude of g is maximizedin suitably engineered miniature systems, as G and x zpf benefit from small cav-ity size and small resonator mass, respectively. Indeed, the highest values of g to date have been achieved in micrometer-size devices such as photonic crystalcavities or disk resonators , with reported values ranging up to about g / π ≈ .In this work, we show that optomechanical coupling rates can be signifi-cantly enhanced by using photonic modes with subwavelength confinement. Werealize a sliced photonic crystal nanobeam in which light is highly confined in ananoscale volume near the moving dielectric interfaces of a low-mass resonator,leading to unprecedented interaction strengths. We use a simple free-space opti-cal setup to address the structure and demonstrate optical tuning of the mechan-ical resonance frequency, as well as sensitive readout of mechanical motion. Theobserved optical forces and measurement sensitivity provide us with two inde-pendent ways to determine the vacuum coupling rate to be g / π ≈ . µ Wof detected power even in a system with modest optical and mechanical qual-ity factors. The operation with a relatively large cavity bandwidth is especiallyattractive for system integration and miniature sensor technologies as well asmeasurement-based control in nano-optomechanical systems.
Working principle.
To realize a large photon-phonon coupling rate g = Gx zpf , we develop a novel system that is based on a patterned silicon photoniccrystal nanobeam, which combines optical confinement with flexural mechanical2otion (Fig. 1). The beam is ‘sliced’ through the middle such that it mechani-cally resembles a pair of doubly clamped beams, coupled through the clampingpoints at the ends of the nanobeam. Figure 1a shows the simulated fundamen-tal in-plane mechanical resonance of the sliced nanobeam structure. The smallwidth (80 nm) of the narrowest parts of the half-beams ensures both the mass( ≈ . x zpf .The in-plane mechanical motion causes a strong change in the separationdistance d . This leads to a giant change of the optical response, as we design thebeam to concentrate light in the subwavelength slit separating the two halves.In general, a displacement-induced frequency shift of an optical mode dependson the fraction of the energy density that is located near the moving dielectricboundaries . To maximize this effect, we rely on the high localization of energythat can occur in systems with dielectric discontinuities with subwavelengthdimensions , in this case provided by the narrow slit through the middle ofthe sliced nanobeam. The periodic patterning of the beam creates a photoniccrystal, with a quasi-bandgap for TE-polarized modes guided by the beam (seeSupplementary Information). The waveguide mode at the lower edge of theband gap has strongly confined electric fields in the nanoscale gap separatingthe ‘teeth’ of the two half-beams (Fig. 1b).The truly subwavelength character of this waveguide is revealed by calculat-ing its effective mode area, which we suitably define as A = (cid:82) d V W ( r ) / ( aW max ),where the energy density W ( r ) = (cid:15) ( r ) | E ( r ) | has its maximum W max just atthe vacuum side of the gap boundary, and we integrate over a full unit cell withperiod a . The mode area is only 2 . × − m for a gap width of 60 nm,or in other words A = 0 . λ , with λ the wavelength in vacuum. In fact, itis even 8 times smaller than the squared wavelength in silicon , even thoughthe maximum energy density is actually localized in the vacuum gap (Fig. 1b).This subwavelength mode area is essential to the sliced nanobeam and makesit stand out with respect to other designs, including the related double-beam‘zipper’ cavity , where the optical cavity modes of two photonic-crystalnanobeams are coupled by placing the beams close together.Recently it was shown that with a similar approach photonic crystal nanobeamcavities can be created that have a high quality factor and an ultrasmall modevolume . We introduce a defect in the periodicity in the middle of the beamso confined cavity modes are created with a frequency in the bandgap. Theseare derived from the band of interest (i.e. the lower bandgap edge) by reducingthe width of the central pair of teeth, such that the effective refractive index islocally reduced. Figure 1c shows the simulated field profile of the lowest-orderoptical cavity mode.Numerical simulations confirm that the frequency of both the band edgeand the defect cavity mode derived from it respond strongly to a displacementof the two half-beams, reaching G = ∂ω c /∂x ≈ π × . . We define thedisplacement coordinate as x = d/
2, such that it can be directly related tothe maximum lab-frame displacement of the antisymmetric mechanical modedepicted in Fig. 1a. Note that the choice of the definition of x is in principlearbitrary (with a properly matched definition of m eff ), whereas the coupling rate g is independent of this definition. To determine the optical frequency shift, the3 d μ m 0max d i s p l a c e m en t a Ω m r=r6rMHz 0 E x max-max ce ω c r=r2 π r×r186.7rTHz 500rnm0 max ε |E| crossrsection 186.7rTHzSi x ̂ μ m1r μ m ∂ ω / ∂ x ( π T H z / n m ) Gap size (nm)
Band edgeCavity mode . .
52 0 30 60 90 120 defect
Figure 1 | Geometry and resonances of the sliced nanobeam. ( a ) Simulateddisplacement profile of the fundamental (in-plane) mechanical resonance of the struc-ture. ( b ) Cross section in the center of the sliced nanobeam (indicated by the dashedline in c ), showing the simulated energy density distribution of the fundamental opti-cal cavity mode of the structure. ( c ) Simulated transverse electric field profile of thefundamental optical cavity mode of the structure. ( d ) Simulated frequency shift as aresult of an outward displacement of 1 nm. The cavity mode shift was determined bysimulating the full nanobeam and introducing a uniform displacement along the beam.( e ) Electron micrograph of a fabricated device. entire half-beams are displaced in the simulation. The displacement of the actualmechanical mode is not uniform along the beam (Fig. 1a), meaning that dueto the finite extent of the optical mode the value of G will be slightly reduced.Taking into account the optical and mechanical mode profiles (Figs. 1a,c), weestimate it to be 0 .
90 times the value shown in Fig. 1d (see SupplementaryInformation).Using standard lithography techniques (see Methods for details), we realizesliced nanobeams in silicon with a length of 11 µ m separated by an average gapsize of 60 nm. An electron micrograph of a fabricated device is shown in Fig. 1e. Free-space readout.
We address our structure using a simple reflectionmeasurement, schematically shown in Fig. 2a. The employed resonant scatteringtechnique places the sample between crossed polarizers to allow the detectionof light scattered by the cavity mode (whose dominant polarization is orientedat 45 ° to the polarizers) while suppressing light reflected by the substrate. Byscanning the frequency of a narrowband laser we record the reflection spectrum,depicted for one of the samples in Fig. 2b. The dispersive lineshape is caused byinterference of the resonant scattering of the cavity with non-resonant scatteringby the nanobeam. The cross-polarized reflectance R is thus well fitted by a Fanolineshape : R (∆) = (cid:12)(cid:12)(cid:12)(cid:12) ce iϕ − √ κ in κ out − i ∆ + κ/ (cid:12)(cid:12)(cid:12)(cid:12) , (1)where c and ϕ are the amplitude and phase of the non-resonant scattering,respectively, and ∆ ≡ ω − ω c is the detuning of the laser frequency ω from thecavity resonance with linewidth κ . The rate at which light can couple to the4avity mode from the free-space input beam is given by κ in , whereas κ out isthe rate at which the cavity decays to the radiation channels that are detectedthrough the output analyzer. In principle, these coupling rates can be unequalbecause the light emitted by the cavity has a spatial mode profile that differsfrom the Gaussian input beam.Fitting equation (1) to the reflection spectrum yields the center frequencyand linewidth of the cavity, as well as a value for κ ex ≡ √ κ in κ out . We determine κ ex ≈ . κ and the optical quality factor Q opt = ω c /κ ≈ Q opt is 2–3 times lower than the simulated one, a discrepancy that we attributeto fabrication imperfections.Thermal motion of the nanobeam δx modulates the cavity frequency by δω c = ( ∂ω c /∂x ) δx . This produces a change in detected power proportionalto the derivative of the reflection spectrum: δP = P in ( ∂R/∂ω c ) δω c . Here weassumed the intracavity amplitude is instantaneously affected by the mechanicalmotion, which is justified since Ω m (cid:28) κ (see Supplementary Information for themore general case). Thus, the power spectral densities of x and P are relatedas S P P (Ω) = P (cid:18) ∂R∂ω c (cid:19) (cid:18) ∂ω c ∂x (cid:19) S xx (Ω) = P (cid:18) ∂R∂ ∆ (cid:19) g x S xx (Ω) . (2)Figure 2c shows the detected spectral density S P P for the laser tuned tothe optical resonance frequency, with a relatively high optical power incidenton the sample ( P in = 367 µ W), corresponding to a detected power of 22 µ W.Because of the linear relation between S P P and S xx shown in equation (2), thisis a direct measurement of the spectrum of thermal motion in the nanobeam.The two peaks at 2 . . d is not affected, and a differentialmode, for which anti-phase movement of the half-beams results in maximalvariation of d . Fabrication-related imperfections can break the symmetry ofthe system, such that the actual normal modes (cid:126)ψ α,β are linear combinationsof the half-beam eigenmodes (cid:126)ψ , : (cid:126)ψ α = A α ( (cid:126)ψ sin θ − (cid:126)ψ cos θ ) and (cid:126)ψ β = A β ( (cid:126)ψ cos θ + (cid:126)ψ sin θ ), where θ can in principle take any value. As we showin the Supplementary Information, the splitting between the mode frequenciesΩ α,β m is enhanced due to the presence of compressive stress in the studied sample,which also reduces the mode frequencies with respect to the simulated value inabsence of stress of 6 MHz. Since the two modes generally affect the separation d differently, they have different photon-phonon coupling rates g , which aremaximal for a purely differential mode ( θ = π/ x = d/ x α zpf /x β zpf = (cid:112) Ω β (1 + sin 2 θ ) / Ω α (1 − sin 2 θ )(see Supplementary Information). The variance in x due to thermal motion inthe two modes is set by the equipartition theorem, taking into account thisdifference in x zpf . The ratio between the areas of the two resonance peaks in theexperimental spectrum of S P P therefore directly yields the mixing angle θ .In fact, fitting two resonant modes to the displacement spectrum also allowsdetermining the transduction factor that relates the measured optical powerspectral density S P P to the displacement spectrum S xx . To do so, we calculate5he thermal variance (cid:104) x (cid:105) th = 2 x k B T / ¯ h Ω m . We determine x zpf from themeasured θ and from the effective mass of purely antisymmetric motion, whichwe computed from the simulated displacement profile to be m eff ≈ . m , with m the total mass of the beam. We further assume that the temperature T of themechanical bath is equal to the lab temperature. The validity of this assumptionis tested by performing power- and detuning-dependent measurements presentedin the Supplementary Information. The resulting scale for the displacementspectral density S xx is shown on the right side of Fig. 2c. Note that the chosenconvention of x allows directly comparing the readout of the two mechanicalresonances on this scale.To determine the sensitivity with which the displacement spectrum of thebeam can be read out, we consider the detection noise floor for the measurementshown in Fig. 2c, which is composed of electronic noise of the photodetector andthe optical shot noise of the detected light. Their measured combined impreci-sion (blue datapoints in Fig. 2c) is over 7 orders of magnitude smaller than themeasured signal.A general assessment of the sensitivity capabilities of the measurement ismade by comparing the detection noise imprecision to the (shot noise) impreci-sion S imp xx (Ω m ) of a resonator read out at the standard quantum limit (SQL) .The imprecision at the SQL is equal to half of the spectral density of the zero-point fluctuations S zpf xx (Ω m ) /
2. We determine this value from the measured ther-mal noise spectrum of the lowest-frequency mode via the average phonon occu-pancy of the mechanical mode k B T / ¯ h Ω m , and indicate it in Fig. 2c with the reddotted line. The optical shot noise of the light impinging on the detector, andeven the total measurement noise floor, are lower than the imprecision noise atthe SQL.Readout of a nanomechanical resonator with an imprecision below that atthe SQL was first achieved in 2009 making use of high-quality optical andmechanical modes. These high quality factors were instrumental because theability to perform a measurement with SQL-level sensitivity scales, per intra-cavity photon, with the single-photon cooperativity C = 4 g /κ Γ. This shows itdepends on the photon-phonon coupling strength as well as the optical linewidth κ = ω c /Q opt and the mechanical linewidth Γ = Ω m /Q m . The fact that here weachieve a detection noise imprecision below that at the SQL with optical and me-chanical quality factors of both less than 500 attests to the large optomechanicalcoupling strength, and could have important application in broadband, sensitivenanoscale sensors. Determining the photon-phonon coupling rate.
To quantify the op-tomechanical interaction strength in the fabricated devices, we model the trans-duction of thermal displacement fluctuations using equation (2) and use it to fita low-power measurement on a structure for various laser detunings. We do thisby calculating the variance of the optical power fluctuations δP at the detectorresulting from displacement fluctuations δx of a mechanical mode with known(thermal) variance. Integrating equation (2) over a single mechanical mode andusing our expression for the reflection spectrum R (∆) (equation (1)), yields (cid:104) P (cid:105) = 8 P g k B T ¯ h Ω m κ (cid:0) ∆ κ ex − c ∆ κ cos( ϕ ) − c (∆ − κ /
4) sin( ϕ ) (cid:1) (∆ + κ / , (3)which is independent of the choice of the displacement coordinate x .6 BSVacuum Low-noisedetectorSample Tunablenarrowband laser
VVH+V H ab c S PP ( W / H z ) S xx ( m / H z ) Frequency (MHz) optical shot noiseelectronic noise max( S ZPF PP )/2noise floor − − − − − . . . . . − − − − − C r o ss - p o l . r e f l e c t i o n ( g ) Optical frequency (THz)
Wavelength(nm) . . . Modulation.spectrum
Figure 2 | Free-space characterisation. ( a ) Schematic diagram of the free-spacereadout method (PBS: polarizing beamsplitter; H,V: horizontally and vertically polar-ized light. See Methods for details). ( b ) Reflection spectrum (red datapoints) and fitwith a Fano lineshape (black line). ( c ) Modulation spectrum of the reflected light ob-tained with the laser frequency on-resonance with the cavity (orange datapoints), anda fit of the two mechanical resonances (brown line). The noise floor (blue datapoints)was obtained by reflecting the laser light from the unpatterned substrate and matchingthe intensity on the detector. The black dashed line is the sum of the measured elec-tronic noise (grey datapoints) and the optical shot noise calculated from the intensityon the detector (light blue dash-dotted line). The red dotted line shows the peak valueof S ZPF PP / S zpf PP (Ω m ) / S th PP (Ω m )¯ h Ω m / k B T . The measured variance of the optically modulated signal due to the lowest-frequency mechanical mode is shown in Fig. 3b. The variance is minimal whenthe derivative of the reflection signal (Fig. 3a) vanishes. Interestingly, due to thedispersive lineshape the transduction is largest for the laser tuned to resonance.The line shown in Fig. 3b is a fit of equation (3) to the data, using only g asa free fitting parameter (all other parameters having been determined in inde-pendent measurements). The corresponding value for g / π is 11 . .To compare this photon-phonon coupling rate to the prediction from oursimulation we estimate the zero-point fluctuations of the structure. Using themeasured mechanical resonance frequency and the simulated effective mass, weobtain x zpf = (cid:112) ¯ h/ m eff Ω m ≈ .
08 pm for a purely anti-symmetric mode. Withthe simulated frequency response G , this yields a prediction of g / π ≈
26 MHz.To take into account the observed asymmetry of the mechanical mode, we shouldapply a correction factor of 0 .
76, based on our knowledge of θ (see Supplemen-tary Information). This results in an expected value of g / π ≈
20 MHz. Weattribute the remaining discrepancy to fabrication imperfections, that could re-sult in a different overlap of the optical and mechanical modes than simulated.So in fact, these simulations show that a further increase of g even beyond themeasured value is possible. Optical spring tuning.
While we tune the laser frequency across the op-tical resonance a pronounced shift of the mechanical resonance frequency is7 r o ss - p o l . r e f l . ( % ) ab S PP ( W / H z ) Frequency(MHz) − − − . . < P > ( W ) Optical frequency (THz) − − − − . . . Figure 3 | Sensitivity of the displacement measurement. ( a ) Reflection spec-trum (red datapoints) and fit with Fano lineshape (black line) for this particularnanobeam. ( b ) Detected optical variance in the fundamental mechanical resonance(blue datapoints) with power incident on the sample P in = 8 . µ W. The datapointswere obtained by fitting the fundamental mechanical resonance peak in the measuredmodulation spectra (inset). The signal originates from thermal motion so it varies onlywith the sensitivity of the measurement. The black line shows our model, which usesthe parameters obtained from the reflection spectrum in a and is fitted to the data todetermine g = 11 . observed. In Fig. 4a this is shown for the same structure we studied in Fig. 3.This well-known optical spring effect is caused by the radiation pressure forcebeing opposed to (aligned with) the mechanical restoring force when the laseris detuned below (above) the resonance frequency, changing the effective springconstant and therefore the mechanical resonance frequency . The equation thatdescribes this behaviour in the limit of a large cavity linewidth ( κ (cid:29) Ω m ) is δ Ω m = g + κ / κ in P in ¯ hω . (4)From equation (4) we recognize that the optical spring tuning shown in Fig. 4provides a second, independent way to characterize the photon-phonon couplingrate. Figure 4b shows the center frequency of the mechanical resonance extractedfrom the same measurement as the variances in Fig. 3b, as well as a fit usingequation (4). To estimate g from this fit we need to know κ in , which we cannoteasily determine as it generally depends on the overlap between the focusedGaussian beam and the cavity mode profile. However, we can find bounds for κ in by considering the total decay rate κ and κ ex = √ κ in κ out , which weredetermined from the fit to the reflection spectrum. On the one hand we know κ in ≤ κ ex , i.e. the collection efficiency is at least as efficient as the overlap witha Gaussian beam, and on the other hand κ in ≥ κ /κ , i.e. at most half of thelight escaping from the cavity can be collected because of the vertical symmetryof the structure. Combining these bounds with the fit of the optical spring effectyields a range for g between 10 and 13 MHz, in good agreement with the valueobtained from the analysis of measurement transduction. The fact that thespring shift can be fully explained by the radiation pressure force as predictedby equation (4) shows that forces due to photothermoelastic effects are likely8 . . . Optical frequency (THz) . . . . . F r e qu e n c y ( M H z ) − − − − − − I n t e n s i t y ( d B m ) F r e qu e n c y ( M H z ) Optical frequency (THz) . . . .
42 207 207 . . . ab Figure 4 | Optical tuning of the mechanical resonance frequency. ( a ) Spectro-gram showing optical tuning of mechanical resonance frequency with power incidenton the sample P in = 140 µ W. ( b ) Fitted frequency of the fundamental mechanicalresonance (blue datapoints) with P in = 8 . µ W. The black line shows a fit using themodel for optical spring tuning (equation (4)). insignificant compared to radiation pressure.
Nonlinear transduction.
As a consequence of the large photon-phononcoupling rate, the thermal motion of the nanobeam induces frequency changesthat are appreciable with respect to the linewidth of the cavity, which results innonlinear transduction. This generates spurious signals at integer multiples of,and combinations of, the strongest modulation frequencies. Detection of suchsignals at multiples of the mechanical resonance frequency resulting from ther-mal motion was reported previously and compared to quadratic optome-chanical coupling .Figure 5a shows a transduced spectrum where we identify 15 peaks as integermultiples and combinations of the two fundamental mechanical resonances at1 . . j,k = | j A ± k B | , with j, k ∈ { , , , . . . } .Peaks corresponding to different order ( j + k ) have a different detuning depen-dence, but all peaks with the same order differ only by a constant factor. Toillustrate the detuning dependence of the higher-order peaks, we plot the vari-ance of the peaks j A for j = { , , , } in Fig. 5b. The detected height of thehigher-order peaks can be predicted by a Taylor expansion of the amount of lightin the cavity around the average detuning (see Supplementary Information),the result of which is shown in Fig. 5c.Note that the higher-order peaks in this calculation were not fitted to thedata, but follow from the value of g we obtained by fitting the first-orderpeak, as shown in Fig. 3. The measured nonlinear sidebands are larger thanexpected (corresponding to a suggested increase of g of about 60%). The originof this discrepancy is unknown. Possible explanations include higher-order op-tomechanical coupling or mechanical nonlinearities . However, the symmetryand shape of the curves match the experimental data, which confirms that thedetuning dependence corresponds to the successive derivatives of the reflection9 B A A A B BA + BB − A A + B A − B B − A A + B A + B C A + B A2AA2A 3A 4A3A 4A abc
THz S PP ( W / H z ) Frequency (MHz) − − − − M e a s u r e d < P > ( W ) − − − − C a l c u l a t e d < P > ( W ) Optical frequency (THz) − − − − − . . . Figure 5 | Nonlinear transduction. ( a ) Measured mechanical spectrum of thesliced nanobeam (purple datapoints), using near-resonant light with power incidenton the sample P in = 8 . µ W. Three peaks A, B and C are mechanical resonances ofthe nanobeam, all other visible peaks correspond to integer multiples and combinationsof frequencies A and B. The electronic noise floor is shown with grey datapoints. ( b )Areas under the peaks corresponding to integer multiples of frequency A, obtained byfitting the peaks in the spectrum. ( c ) Calculated variance of the reflected signal, usingthe experimentally obtained parameters. spectrum (Fig. 3a). Discussion.
The free-space readout method we employ provides an easyand robust way of coupling light to the cavity. We have intentionally engineeredthe cavity defect such that it has a significant dipole moment , allowing cou-pling to free space at an appreciable rate. This makes it unnecessary to createan explicit loss channel for coupling, e.g. in the form of a grating or feedingwaveguide. Moreover, the Fano-shape of the reflection spectrum allows directtransduction of motion to optical amplitude modulation for a laser tuned tothe cavity resonance (where dynamical radiation pressure backaction is zero),without more complicated interferometric schemes. As a result of the efficientcoupling to free space, the bandwidth of the cavity is large (0.5 THz), whichis appealing in the context of applications that require frequency matching ofmultiple systems: together with the small system footprint, it could assist theintegration of such optomechanical transducers in sensor arrays or effectiveoptomechanical metamaterials .Of course, for applications that benefit from enhanced measurement sensi-tivity such as measurement-based control of the mechanical quantum state, itcould be worthwhile to realize a higher optical quality factor by introducingtapering along the nanobeam . To simultaneously allow efficient free-spacecoupling would in such a case require special attention, in the form of tailoringthe spatial mode profile of the cavity radiation. This could be especially im-portant for effects that depend on the intracavity photon number, such as the10emonstrated optical spring effect, as the rates κ in and κ out will differ. Furtherquantification of their individual magnitudes (e.g. through systematic variationof incident and detected mode profiles) will thus be valuable.Likewise, we expect that the mechanical quality factor for the nanobeams weemploy can be improved with suitable design principles and optimization of thefabrication process. Indeed, measurements on similar-sized silicon nanobeamsand cantilevers suggest that quality factors in the range of 10 to 10 shouldbe possible at room and cryogenic temperature, respectively . Nonetheless,we point out that because of our large coupling rate, even with the currentmodest values of both optical and mechanical quality factors the single-photoncooperativity in this structure reaches C = 0 .
16. The value of this quantity,which compares optomechanical coupling strength and dissipation, and is forexample a measure for the capability of the system to perform measurementsat the SQL, is on par with many recently reported systems with much higherquality factors .In conclusion, we demonstrated an optomechanical device with a large photon-phonon coupling rate g / π = 11 . g exceeds the mechan-ical resonance frequency Ω m , which is one of the requirements for ultrastrongcoupling . With further improvements in both the coupling rate and theoptical quality factor, the present approach might provide a route to simultane-ously reach g > Ω m and g ≈ κ . It will be interesting to explore to what extentthis regime can be used to exploit nonlinear optomechanical interactions at thesingle-photon level. Acknowledgements.
The authors thank R. Thijssen for valuable discus-sions. This work is part of the research programme of the Foundation for Funda-mental Research on Matter (FOM), which is part of the Netherlands Organisa-tion for Scientific Research (NWO). E.V. gratefully acknowledges an NWO-Vidigrant for financial support.
Methods
Numerical simulation.
All numerical eigenmode simulations were performedusing finite-element software COMSOL Multiphysics. In mechanical simulations,the connection between the substrate and the support pads was modeled as afixed boundary, while all other boundaries were kept free. To find the guidedmodes of the photonic crystal nanobeam, a unit cell was simulated with Floquetboundary conditions along the propagation direction and in the other directionsperfect electric conductors at several micrometers distance from the structure.Finally, to simulate the cavity mode, a full nanobeam including support pad wasmodeled with perfectly matched layers on all sides, again at several micrometersdistance from the beam. 11 abrication.
The structures were fabricated from a silicon-on-insulatorwafer with a device layer thickness of 200 nm, and a buried oxide layer of 1 µ mthick. A resist layer of hydrogen silsesquioxane (HSQ) with a thickness of 80 nmwas spincoated on top and patterned using electrons accelerated with 30 kV.The resist was developed using TMAH and then the pattern was transferredto the silicon layer using a reactive-ion etch process with SF /O gases, opti-mized for anisotropy and selectivity. To release the structures, the oxide layerwas dissolved in a 20% HF solution. After this step the structure was driedwith a critical point dryer to prevent the sliced beams being pulled togetherduring the drying process. The suspension of the nanobeams from their supportpads was designed to allow some relief of compressive stress along the beam.The compressive stress is present in most free-standing structures created fromSOI , but it has a large effect for our structures because of their low stiffness. Free-space setup.
The laser beam (New Focus Velocity 6725) was focusedon the sample by an aspheric lens with a numerical aperture of 0 .
6. A polarizingbeamsplitter provided a cross-polarized detection scheme, where any light thatwas directly reflected was rejected and only light that coupled to the sample,placed at 45 ° , was transmitted to the detector. Both the lens and the samplewere in a vacuum chamber to reduce mechanical damping by air molecules. Allexperimental results shown were performed with a pressure of about 4 mbar,except the spectra in Fig. 2, where the pressure was lower than 10 − mbar. At apressure of 4 mbar, the mechanical quality factor was lowered to approximately200. Analysis of modulated reflection signals.
We detected the reflectionsignal using a low-noise InGaAs-based photoreceiver (Femto HCA-S) and an-alyzed it using an electronic spectrum analyzer (Agilent MXA). We fitted thepeaks in the modulation spectra using a Lorentzian convolved with a Gaussiandistribution, also called a Voigt lineshape. The Gaussian contribution accountedfor the resolution bandwidth of the spectrum analyser, as well as for frequency-noise broadening at relatively large optical input power P in . With high P in , smallfluctuations in incoupling efficiency or laser intensity thermally shifted the cav-ity resonance, which resulted in frequency noise via optical spring tuning of themechanical resonance. References
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Supplementary InformationModel for transduction
We derive the effect of a small frequency modulation of the intracavity field atthe output of a cavity, which allows us to set up a model that describes the trans-duction of thermal motion to optical intensity modulation. This derivation fol-lows the calculation shown by Gorodetsky, Schliesser, Anetsberger, Deleglise &Kippenberg [1], with the important difference that we include the non-resonantcontribution in the reflection spectrum that leads to the Fano lineshape weobserve.We start from the equations describing the behaviour of the optomechanicalsystem ˙ a = ( i (∆ − Gx ( t )) − κ/ a ( t ) + √ κ in s in ( t ) , ¨ x ( t ) + Γ m ˙ x ( t ) + Ω x ( t ) = − ¯ hG | ¯ a | , (S1)where a is the internal field in the cavity, s in is the input field related to theinput power P in = ¯ hω | s in | , ∆ is the detuning of the input light from the cavityresonance ω c , κ is the cavity decay rate, κ in is the coupling rate to the inputchannel and G = ∂ω c /∂x is the optomechanical frequency response. The fre-quency Ω m and damping rate Γ m of the mechanical resonator are influenced bythe number of photons in the cavity | ¯ a | , an effect we neglect in the followingby assuming a low input power. This simplification is motivated by the factthat we seek to predict the amplitude of the mechanically-induced light modu-lation, not the frequency of such modulations. Moreover, dynamical backactionaffecting mechanical linewidth is small in devices that have large κ/ Ω m .We consider a small harmonic oscillation of the mechanical resonator x ( t ) = x cos(Ω m t ), which causes a modulation of the cavity frequency with amplitude x G , or a modulation of the optical intracavity phase with amplitude x G/ Ω m .15f the modulation is small ( x G (cid:28) κ ), this yields a x = s in √ κ in L (0) × (cid:18) − i x G L (Ω m ) e − i Ω m t − i x G L ( − Ω m ) e i Ω m t (cid:19) , L (Ω) = 1 − i (∆ + Ω) + κ/ . (S2)The cavity is coupled to the output s out with the coupling rate κ out . The equa-tion for the output field reads s out = ce iϕ s in − √ κ out a x , (S3)where the first term is caused by the nonresonant scattering from the input tothe output with amplitude c and phase ϕ .In the experiment, we measure the intensity | s out | and feed it to a spectrumanalyser, which yields the single-sided spectrum of the signal. Since we areinterested in the strength of the spectral component at the mechanical oscillationfrequency Ω m , we highlight the time dependence here: | s out | ( t ) = c | s in | + κ out | a x ( t ) | − c √ κ out ( e iϕ s in a ∗ x ( t ) + e − iϕ s ∗ in a x ( t )) . (S4)The first term is constant so it does not contribute to a signal at Ω m . Substi-tuting a x into the other two terms, and discarding any terms not oscillating at ± Ω m yields | s out | ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ± Ω m = 12 x G | s in | √ κ in κ out × (cid:18) √ κ in κ out | L (0) | (cid:0)(cid:2) ie i Ω m t ( L ∗ (Ω m ) − L ( − Ω m )) (cid:3) + c . c . (cid:1) − c (cid:0)(cid:2) ie i Ω m t (cid:0) e iϕ L ∗ (0) L ∗ (Ω m ) − e − iϕ L (0) L ( − Ω m ) (cid:1)(cid:3) + c . c . (cid:1) (cid:19) . (S5)This expression contains the modulation amplitude. In our experiment, we com-pare the variance of the modulation to the known variance of the mechanicalthermal motion (cid:104) x (cid:105) th = 2 x k B T / ¯ h Ω m . Therefore we calculate the varianceof P out = ¯ hω | s out | due to the modulation at +Ω m and − Ω m , which will bothcontribute to the signal at +Ω m in the single-sided spectrum. We can write P out (cid:12)(cid:12)(cid:12) ± Ω m = Ae i Ω m t + A ∗ e − i Ω m t , which leads to (cid:104)| P out | (cid:105) Ω m = 2 | A | . After somealgebra, we arrive at (cid:104)| P out | (cid:105) Ω m = 2 x G P in ( κ in κ out )(∆ + κ / ((∆ + Ω m ) + κ / − Ω m ) + κ / × (cid:104) ∆ κ in κ out − c √ κ in κ out (∆ κ cos ϕ + (∆ − κ /
4) sin ϕ )+ c (cid:18) ∆ (Ω + κ ) cos ( ϕ )+ (∆ − ∆ κ / κ Ω / κ /
16) sin ( ϕ ) − ϕ ) sin( ϕ )( − ∆ κ + ∆ κ / κ Ω / (cid:19)(cid:105) . (S6)16f we evaluate this expression in the bad-cavity limit (Ω m (cid:28) κ ), we find itis directly related to the derivative of the Fano lineshape ∂R/∂ ∆: (cid:104)| P out | (cid:105) Ω m = 2 x G P κ in κ out (cid:0) ∆ √ κ in κ out − c ∆ κ cos( ϕ ) − c (∆ − κ /
4) sin( ϕ ) (cid:1) (∆ + κ / = 12 x G P (cid:18) ∂R∂ ∆ (cid:19) . (S7)We note that imperfect transmission of the optics between the sample andthe detector scales the detected signal in the same way as the input power P in ,and will enter the equations in the same way. Finally, we substitute the variancedue to the modulation amplitude x by the variance of the mechanical motion: x G → (cid:104) x (cid:105) th G = 4 g k B T / ¯ h Ω m , which leads to equation 3 in the main text. Nonlinear transduction
The previous section started from the assumption that the frequency modula-tion δω c = Gδx is small with respect to the cavity linewidth, δω c (cid:28) κ , andconsidered only the resulting linear transduction at the modulation frequencyΩ m . In this section we show that the first signature of large δω c is the ap-pearance of nonlinear transduction, which produces a signal at multiples of themodulation frequency Ω m . For the second-order transduction, this was shownby Doolin et al. [2], where also a quadratic optomechanical coupling was takeninto account. Here we derive the result for any higher-order terms of nonlineartransduction.In the non-resolved sideband regime ( κ (cid:29) Ω m ), the optical fields in the cavityreach a steady state much faster than the timescale of mechanical motion. Theintracavity amplitude can then be written as a ( t ) = √ κ in s in − i (∆ − δω c ( t )) + κ/ , (S8)which combined with equation (S3) yields | s out | | s in | = c + 4 κ in κ out κ
11 + u − c √ κ in κ out κ e iϕ (1 − iu ) + e − iϕ (1 + iu )1 + u . (S9)Here we defined u ≡ − δω c ( t )) /κ , which implies u is detuning- and time-dependent. We now summarize equation (S9) as R (cid:48) ( u ) and find the Taylor ex-pansion for small δω c around u ≡ /κ : R (cid:48) ( u ) = R (cid:48) ( u ) − δω c κ ∂R (cid:48) ( u ) ∂u + . . . + ( − δω c /κ ) n n ! ∂ n R (cid:48) ( u ) ∂u n , (S10)where the last term depicts the n th order in the Taylor expansion. We takea harmonic modulation of the cavity frequency δω c ( t ) = A cos Ω t . To leadingorder, δω n c ≈ A n cos( n Ω t ) / n − . This means that each successive term in theTaylor expansion in equation (S10) gives the amplitude of a term at differentfrequency.In the optomechanical system, the variance of the frequency modulations atthe mechanical frequency Ω m is given by (cid:104) δω (cid:105) = G (cid:104) x (cid:105) th = 2 g k B T / ¯ h Ω m .17herefore we get the same variance in R if we set A = 2 g (cid:112) k B T / ¯ h Ω m . We notethat ∂ n R (cid:48) ( u ) ∂u n = ( κ/ n ∂ n R (∆) ∂ ∆ n , which for the variance of the signal at n Ω m leadsto (cid:104) P (cid:105) P (cid:12)(cid:12)(cid:12)(cid:12) n Ω m = (cid:104) R (cid:105) (cid:12)(cid:12)(cid:12) n Ω m = 2( g k B T / ¯ h Ω m ) n n ! (cid:18) ∂ n R (∆) ∂ ∆ n (cid:19) . (S11)For n = 1, the result of equation 3 in the main text is again reproduced. Theresult of a calculation of the variances of n = 1–4, for the parameters used inthe experiment, is shown in the main text in Fig. 5c. Waveguide modes in the sliced nanobeam
In this section we discuss the waveguide modes in the periodic region of thesliced nanobeam in more detail. The free-standing silicon nanobeam acts as awaveguide, which can guide light via total internal reflection. In this waveguide,the elliptical holes form a photonic crystal that opens a bandgap for modeswith transverse electric (TE)-like symmetry (Supplementary Figure S1a). Thisis not a full bandgap, since TM-like waveguide modes exist in the gap region.If the symmetry of the structure is broken by fabrication imperfections, lightin the bandgap region for TE-like modes can scatter to the TM-like modes andpropagate along the nanobeam. For this reason, this is sometimes referred to asa quasi-bandgap.The guided mode at the lower edge of the band gap has the largest concen-tration of energy in the nanoscale gap in the middle of the beam. The fact thata significant portion of this mode’s energy is located in vacuum increases its fre-quency in comparison to a non-sliced nanobeam, which reduces the frequencywidth of the bandgap. To ensure maximum mirror strength, the transverse sizeof the holes is made as large as possible. The elliptical hole shape is as suchimportant to realize a strong bandgap, in addition to providing favorable me-chanical properties as mentioned in the main text.To create optical cavity modes that are derived from the lower band edge,the defect is a local decrease in distance between two elliptical holes, whichdecreases the local effective refractive index and creates defect states in thebandgap region. Supplementary Figure S1b shows the first two cavity modescreated in this way in the sliced nanobeam. Note that the higher-order cavitymodes have a lower frequency, since they are less confined near the defect, so thatthe frequency is closer to that of the waveguide mode in the periodic structure.
Overlap of optical and mechanical mode profiles
The frequency shift of the optical resonance of the sliced nanobeam due to me-chanical motion depends on the overlap between the optical and mechanicalmode profiles. We simulated the frequency shift of the optical resonance of thesliced nanobeam due to a uniform mechanical shift of 1 nm, as shown in themain text in Fig. 11d. To estimate the influence of the finite extent of the modeprofiles on the response, we extract these profiles from the numerical simulation.Supplementary Figure S2 shows the normalized displacement profile of the me-chanical resonance and the normalized electromagnetic energy density profile ofthe optical cavity mode, as a function of the position along the nanobeam.18 b μ m x max-max defect x ̂ ω c = 2 π × 186.7 THz ω c = 2 π × 183.0 THz O p t i c a l f r e qu e n c y ( T H z ) k ∥ ( π / a ) li gh t li ne . . . . Figure S1 | Optical modes in the sliced nanobeam. ( a ) Simulated disper-sion diagram showing the TE-like waveguide modes in the periodic part of the slicednanobeam. A bandgap is opened by the periodic structure of elliptical holes around afrequency of 200 THz. ( b ) Simulated transverse electric field profiles of the first twocavity modes of the structure. The defect that is responsible for the creation of thesemodes is a slightly smaller distance between the two holes in the center of the beam. The mechanical mode profile closely resembles the mode profile of the fun-damental mode of a doubly-clamped beam. From the simulated displacementprofile we calculate the effective mass for this purely antisymmetric motion tobe m eff ≈ . m , where m is the total mass of the sliced nanobeam, which isindeed very close to the value obtained from the analytical displacement profileof a uniform doubly-clamped beam[3].The optical mode profile clearly shows the localization of the energy densityin the small gaps between the silicon ‘teeth’ of the structure. The simple defectwe introduce in the center of the beam localizes the optical cavity mode there,while the field decays exponentially away from the defect, where the opticalfrequency lies inside the photonic quasi-bandgap.We compute the correction on the frequency shift due to the finite extent ofthe modes from the overlap integral between these two mode profiles, and finda factor of 0 .
90 with respect to the frequency shift for a uniform displacementof the beam.The higher-order cavity modes created by the defect are less strongly con-fined along the length of the beam, as shown in Supplementary Figure S1b.Since the fundamental mechanical resonance has the largest displacement inthe center of the beam, the higher-order cavity modes are less sensitive to thismotion than the fundamental optical resonance.
Mechanical mode coupling
The mechanical modes observed in a two-beam system are combinations of themotion of the individual beams, with in the perfectly symmetrical case fullyin-phase or out-of-phase motion. Here we derive the consequences of imperfectsymmetry for the ratio of scattered power between the two modes[4].Harmonic motion of the two beams at a certain frequency Ω can be describedas: (cid:20) x ( t ) x ( t ) (cid:21) = (cid:20) ψ ψ (cid:21) cos Ω t ≡ (cid:126)ψ cos Ω t , where (cid:126)ψ = (cid:20) ψ ψ (cid:21) . (S12)Thus, ψ and ψ are the amplitudes of the oscillatory motion of the two indi-19 ( µm ) D i s p l ace m e n t ( a . u . ) E n e r gyd e n s it y ( a . u . ) Figure S2 | Normalized resonant mode profiles of the fundamental optical and me-chanical resonance of the sliced nanobeam. The red dotted line shows the displacementalong the length of the beam, while the blue solid line represents the local energy den-sity. vidual beams, such that their variance is (cid:104) ψ , (cid:105) = ψ , .The optical response of the system is determined by the change in the dis-tance between the beams: d = x − x . Here we define x as x ≡ d/
2, which leadsto the same value of G = ∂ω c /∂x for both mechanical modes. Note that thischoice of x corresponds to the lab-frame displacement of the two beams if theymove in antiphase. The variance of x due to harmonic motion described by (cid:126)ψ isthen: (cid:104) x (cid:105) ψ = (cid:0) ψ + ψ − ψ ψ (cid:1) . (S13)The state vectors of the two normal modes can be written without loss ofgenerality as (cid:126)ψ α = A α (cid:20) cos θ sin θ (cid:21) , (cid:126)ψ β = A β (cid:20) sin θ − cos θ (cid:21) . (S14)For both of these modes, we can calculate the variance of x , denoted as (cid:104) x (cid:105) α and (cid:104) x (cid:105) β , respectively: (cid:104) x (cid:105) α = 18 A α (cos θ + sin θ + 2 sin θ cos θ )= A α θ ) , (cid:104) x (cid:105) β = A β − sin 2 θ ) (S15)For the beams undergoing thermal motion, the variance is given by theequipartition theorem: (cid:104) x (cid:105) α = k B Tm α Ω α , (cid:104) x (cid:105) β = k B Tm β Ω β , (S16)where m α and m β are the effective mass of these modes. As shown in theprevious section, for the differential mode the simulated effective mass withrespect to the displacement coordinate x is m eff = 0 . m , with m the total massof the sliced nanobeam. Evaluating equations (S15) and (S16) for a differentialmode ( θ = π/
4) yields A α = 4 k B T /m eff Ω α and similarly for A β . Substituting20his back into equation (S15), we arrive at (cid:104) x (cid:105) α = k B T (1 + sin 2 θ ) m eff Ω α , (cid:104) x (cid:105) β = k B T (1 − sin 2 θ ) m eff Ω β . (S17)We note that thermal variance is related to the zero-point fluctuations x zpf as (cid:104) x (cid:105) ψ = 2 k B T ¯ h Ω ψ ( x ψ zpf ) , so x ψ zpf = (cid:115) ¯ h (1 ± sin 2 θ )4 m eff Ω ψ , (S18)where ψ = α, β and + respectively − is chosen as the sign for the term sin 2 θ .Finally, we measure (cid:104) P (cid:105) and wish to relate this to a displacement vari-ance (cid:104) x (cid:105) . We calculated m eff using the simulated mode profile and assume thethermal bath temperature of the mechanical modes T is equal to the lab tem-perature, which leaves only θ and the transduction factor ∂P/∂x unknown. Bymeasuring the area of both peaks (cid:104) P (cid:105) α and (cid:104) P (cid:105) β , we resolve the remainingambiguity, allowing us to calibrate the displacement spectrum. Influence of compressive stress and experimentaldisorder
The simulation of the ideal structure shown in the main text predicts the res-onance frequency of the fundamental in-plane resonance to be 6 MHz, and ad-ditionally the frequency difference between the anti-symmetric and symmetricmode is negligible. We expect the frequency of the out-of-plane resonances tobe larger than that of the fundamental in-plane mode, as the narrowest part ofthe half-beams (80 nm) is smaller than the Si slab thickness (200 nm).In our experiment we find significantly smaller values of 2 . . . . .
09% of the length ofthe beam), the resonances occur at frequencies very close to the experimentallymeasured values, at 1 . . max d i s p l a c e m en t Ω m/2 π =1.5 MHz Ω m/2 π =1.9 MHz μ m Figure S3 | Simulation including disorder and compressive stress.
Simulatedmechanical displacement profiles of the two fundamental in-plane resonances. Thedimensions of the beam were matched to the realized dimensions using measurementswith a scanning electron microscope, including differences in hole and gap size along thebeam. In addition, a compressive stress was introduced in the simulation by displacingone of the support pads by 10 nm along the direction of the beam.
Influence of optical input power
In the measurements shown in the main text, we use up to 370 µ W of opticalpower incident on the nanobeam. Using the parameters of our fit to the re-flection spectrum, we estimate that this results in an intracavity intensity thatcorresponds to a maximum of ≈ References
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Opt. Express et al.
Nonlinear optomechanics in the stationary regime.
Phys.Rev. A P > ( W ) P in (W) ∝ P in − − − − − − − − Figure S4 | Power dependence.
The red and blue datapoints show the measuredsignal from the two fundamental mechanical resonances, obtained by fitting both me-chanical resonances in the spectra (see Fig. 2 in the main text for such a spectrum).The data was taken at various input powers with resonant laser light (zero detun-ing). The green line is a guide to the eye, with a slope corresponding to quadraticdependence on input power. The errorbars indicate readout error of the input powerbut don’t take into account possible variations in incoupling efficiency due to slightchanges in alignment.
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