Optical detection of radio waves through a nanomechanical transducer

Optical detection of radio waves through a nanomechanical transducer
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  Optical detection of radio waves through a nanomechanical transducer T. Bagci 1 , A. Simonsen 1 , S. Schmid 2 , L. G. Villanueva 2 , E. Zeuthen 1 , J. Appel 1 ,J. M. Taylor 3 , A. Sørensen 1 , K. Usami 1 , A. Schliesser 1 , ∗ and E. S. Polzik 1 , † 1 Niels Bohr Institute, University of Copenhagen, Denmark  2 Department of Micro– and Nanotechnology, Technical University of Denmark,DTU Nanotech, 2800 Kongens Lyngby, Denmark and  3 Joint Quantum Institute/NIST, College Park, Maryland, USA Low-loss transmission and sensitive recovery of weak radio-frequency (rf) and mi-crowave signals is an ubiquitous technological challenge, crucial in fields as diverse asradio astronomy, medical imaging, navigation and communication, including those of quantum states. Efficient upconversion of rf-signals to an optical carrier would al-low transmitting them via optical fibers instead of copper wires dramatically reducinglosses, and give access to the mature toolbox of quantum optical techniques, routinelyenabling quantum-limited signal detection. Research in the field of cavity optomechan-ics [1, 2] has shown that nanomechanical oscillators can couple very strongly to eithermicrowave [3–5] or optical fields [6, 7]. An oscillator accommodating both these func-tionalities would bear great promise as the intermediate platform in a radio-to-opticaltransduction cascade. Here, we demonstrate such an opto-electro-mechanical trans-ducer following a recent proposal [8] utilizing a high-Q nanomembrane. A moderatevoltage bias ( V  dc  <  10V ) is sufficient to induce strong coupling [4, 6, 7] between thevoltage fluctuations in a radio-frequency resonance circuit and the membrane’s dis-placement, which is  simultaneously   coupled to light reflected off its metallized surface.The circuit acts as an antenna; the voltage signals it induces are detected as an opticalphase shift with quantum-limited sensitivity. The corresponding half-wave voltage isin the microvolt range, orders of magnitude below that of standard optical modulators.The noise added by the mechanical interface is suppressed by the electro-mechanicalcooperativity  C  em  ≈  6800  and has a temperature of   T  N  =  T  m /C  em  ≈  40mK , where  T  m  isthe room temperature at which the entire device is operated. This corresponds to asensitivity limit as low as  5pV / √  Hz , or − 210dBm / Hz  in a narrow frequency band around 1MHz . Our work introduces an entirely new approach to all-optical, ultralow-noise de-tection of classical electronic signals, and sets the stage for coherent upconversion of low-frequency quantum signals to the optical domain [8–12]. Opto- and electromechanical systems [1, 2] have gainedconsiderable attention recently for their potential as hy-brid transducers between otherwise incompatible (quan-tum) systems, such as photonic, electronic, and spin de-grees of freedom [10, 13, 14]. Coupling of radio-frequencyor microwave signals to optical fields via mechanics isparticularly attractive for today’s, and future quantumtechnologies. Photon-phonon transfer protocols viableall the way to the quantum regime have already been im-plemented in both radio- and optical-frequency domainsseparately [5, 7, 15].Among the optomechanical systems that have beenconsidered for radio-to-optical transduction [8, 10, 12,16], we choose an approach [8] based on a very high Q m  ≈  3 · 10 5 nanomembrane [17, 18] which is coupledcapacitively [19–21] to a radio-frequency (rf) resonancecircuit, see Figure 1.Together with a four-segment gold electrode, the mem-brane forms a capacitor, whose capacitance depends onthe membrane-electrode distance  d  +  x . With a tuning ∗ Electronic address: † Electronic address: capacitor  C  0 , the total capacitance  C  ( x ) =  C  0  +  C  m ( x )forms a resonance circuit with a typical quality factor Q LC   =   L/C/R  of 130 when an inductor wired on alow-loss ferrite rod ( L  = 0 . 64mH) is used. The inductorserves as an antenna feeding rf signals into the series  LC  -circuit. The circuit’s resonance frequency Ω LC   = 1 / √  LC  is tuned to the frequency Ω m / 2 π  = 0 . 72MHz of the fun-damental drum mode of the membrane. The membrane-circuit system is coupled to a propagating optical modereflected off the membrane.The electromechanical dynamics is described mostgenerically by the Hamiltonian [8] H   =  φ 2 2 L  +  p 2 2 m  +  m Ω 2 m x 2 2 +  q  2 2 C  ( x )  − qV  dc  (1)where  φ  and  q  , the flux in the inductor and the chargeon the capacitors, are conjugate variables for the  LC  circuit;  x  and  p  denote the position and momentum of the membrane with an effective mass  m . The last twoterms represent the charging energy  U  C  ( x ) of the capac-itors, which can be offset by an externally applied biasvoltage  V  dc  (Fig. 1). This energy corresponding to thecharge ¯ q   =  V  dc C  (¯ x ) leads to a new equilibrium position ¯ x of the membrane. Furthermore, the position-dependentcapacitive force  F  C  ( x ) =  − dU  C dx  causes spring soften-   a  r   X   i  v  :   1   3   0   7 .   3   4   6   7  v   2   [  p   h  y  s   i  c  s .  o  p   t   i  c  s   ]   2   A  u  g   2   0   1   3  2 FIG. 1:  Optoelectromechanical system.  The centralpart of the optoelectromechanical transducer ( a ) is an Al-coated SiN 500 µ m square membrane in vacuum ( <  10 − 5 mbar). It forms a position-dependant capacitor  C  m ( x  = 0) ≈ 0 . 5pF with a planar 4-segment gold electrode in the immedi-ate vicinity (0 . 9 µ m  < ∼ d < ∼ 6 µ m). A laser beam is reflected off the membrane’s Al coating [22], converting its displacementinto a phase shift of the reflected beam. ( b ) The membranecapacitor is part of a resonant  LC  -circuit, tuned to the me-chanical resonance frequency by means of a tuning capacitor C  0  ≈ 80pF (see SI for details). A bias voltage  V  dc  applied tothe capacitor then couples the excitations of the  LC  -circuitto the membrane’s motion. The circuit is driven by a voltage V  s  in series, which can be injected through the indicated cou-pling port ‘2’ or picked up by the inductor from the ambientrf radiation. ( c ) For a membrane-electrode distance of 0 . 9 µ m,the optically observed response of the membrane to a weakexcitation of the system clearly shows a split peak (dashedlines: fitted Lorentzian resonances), due to hybridisation of the  LC  -circuit mode with the mechanical resonance of themembrane. ing, reducing the membrane’s motional eigenfrequencyby ∆Ω m  ≈− C  ′′ (¯ x ) V  2dc / 2 m Ω m  [19].Much richer dynamics than this shift may be expectedfrom the mutually coupled system (1). For small ex-cursions ( δq,δx ) around the equilibrium (¯ q, ¯ x ), it can bedescribed by the linearised interaction term (Ref. [8] andSI) H  I  =  Gδqδx  = ¯ hg em δq    ¯ h/ 2 L Ω LC  δx   ¯ h/ 2 m Ω m ,  (2)parametrized either by the coupling parameter  G  = − V  dc C  ′ (¯ x ) C  (¯ x )  or the electromechanical coupling energy¯ hg em . This coupling leads to an exchange of energybetween the electronic and mechanical subsystems atthe rate  g em ; if this rate exceeds their dissipation ratesΓ LC   = Ω LC  /Q LC  , Γ m  = Ω m /Q m , they hybridise into astrongly coupled electromechanical system [4, 6, 7].Our system is deeply in the strong coupling regime(2 g em  = 2 π  · 36kHz  >  Γ LC   ≫  Γ m  for a distance  d  =0 . 9 µ m and a bias voltage of   V  dc  = 16 . 4V (Fig. 1c). Here,for the first time, we detect the strong coupling using anindependent optical probe on the mechanical system.We have performed an experimental series, in whichthe bias voltage is systematically increased, with a dif-ferent sample and a larger distance  d  = 5 . 5 µ m. Thesystem is excited inductively through port ‘2’ (Fig. 1c),inducing a weak radio wave signal of (r.m.s) amplitude V  s  = 670nV, at a frequency Ω  ≈  Ω LC  . The response of the coupled system can be measured both electrically asthe voltage across the capacitors (port ‘1’ in Fig. 1b) andoptically by analyzing the phase shift of a light beam(wavelength  λ  = 633nm) reflected off the membrane.Both signals are recorded with a lock-in amplifier, whichalso provides the srcinal excitation signal.The electrically measured response (Fig. 2a) clearlyshows the signature of a mechanically induced trans-parency [23–25] indicated by the dip in the  LC   resonancecurve. Independently, we observe the rf signal generatedin the  LC   circuit optically via the membrane mechanicaldynamics (Fig. 2b). In particular, the electromechani-cal coupling leads to broadening of the mechanical reso- FIG. 2:  Mechanically induced transparency.  Re-sponse of the coupled system to weak excitation (throughport ‘2’ in Fig. 1b) probed though both, ( a ) the voltagemodulation in the  LC   circuit (at port ‘1’), and ( b ) the op-tical phase shift induced by membrane displacements. Thedata (coloured dots) measured for five different bias volt-ages agree excellently with model fits (curves) correspondingto  g em / 2 π  =  { 280 , 470 , 810 , 1030 , 1290 } Hz. Note that eachcurve is offset so that its baseline corresponds to the appliedbias indicated on the scale between the panels. Grey circles in-dicate the mechanical resonance frequency extracted for eachset of data. A shift ∆Ω m  ∝ − V  2DC  can clearly be discerned(dashed line is a fit). The inset shows the effective linewidth of the mechanical resonance extracted from full model fits to theelectrically (circles) and optically (boxes) measured responseand simple Lorentzian fits to the optical data (diamonds).  3nance, an electromechanical damping effect analogous tooptomechanical dynamical backaction cooling [26–28], toa new effective linewidth Γ eff   = (1+ C em ) · Γ m , where C em is the electromechanical cooperativity C em  = 4 g 2em Γ m Γ LC  .  (3)The width of the induced transparency dip and the me-chanical linewidth grow in unison, and in agreement withour expectations as Γ eff   ∝  V  2dc  (inset). Both these fea-tures also shift to lower frequencies as the bias voltageis increased, following the expected ∆Ω m  ∝− V  2dc  depen-dence [19]. Note that in each experiment we have tunedthe  LC   resonance frequency to Ω m .Using the model based on the full Langevin equations(SI), derived from the Hamiltonian (1), we fit the elec-tronically and optically measured curves, and obtain fitparameters Ω m , Ω LC , Γ LC  , and  G  which for the twocurves agree typically within 1%. Together with the in-trinsic damping Γ m / 2 π  = 2 . 3Hz determined indepen-dently from thermally driven spectra, the system’s dy-namics can be quantitatively predicted. Our data anal-ysis allows us to quantify the coupling strength in threeindependent ways, by (i) analysis of the mechanical re-sponses’ spectral shape, (ii) comparison of the voltageand displacement modulation amplitudes, and (iii) thefrequency shift [19] of the mechanical mode, and com-pare these experimental values with (iv) a theoreticalestimate taking the geometry of the electromechanicaltransducer into account (SI). For  V  dc  = 125 V we find G  = 10 . 3kV / m following to the first method, and sim-ilar values using the three others (cf. SI), corroboratingour thorough understanding of the system.In another experimental run ( d  = 4 . 5 µ m, Fig. 3), wehave characterised the strong electromechanical coupling[3–5] via the normal mode splitting [29, 30] giving rise to an avoided crossing of the resonances of the electroniccircuit and the mechanical mode, as the latter is tunedthrough the former using the capacitive spring effect [19].In contrast to earlier observations [4, 6, 7] we can si-multaneously witness the strong coupling through theoptical readout, in which the recorded light phase re-produces the membrane motion (Fig. 3c,e). Again, thepredictions derived from the Langevin equations are inexcellent agreement with our observations, yielding a co-operativity of  C em  = 3800 for these data with  m  = 24ng,Γ m / 2 π  = 3 . 1Hz.We now turn to the performance of this interface asan rf-to-optical transducer. A relevant figure of meritfor the purpose of bringing small signals onto an opti-cal carrier is the voltage  V  π  required at the input of theseries circuit in order to induce an optical phase shiftof   π . Achieving minimal  V  π  requires a tradeoff betweenstrong coupling and induced mechanical damping. Forthe optimal cooperativity  C em  = 1 we find V  π  = 12   mL Γ m Γ LC  λ Ω r  ≈ 140 µ V ,  (4) FIG. 3:  Strong coupling regime.  ( a ) Measured coherentcoupling rate 2 g em / 2 π  as a function of bias voltage (points)and linear fit (line). The shaded area indicates the dissipationrate Γ LC / 2 π  ≈  5 . 9kHz of the LC circuit. ( b - e ) Normalisedresponse of the coupled system as measured on port ‘1’ (Fig.1c) ( b,d ) and via the optical phase shift induced by mem-brane displacements ( c,e ). Upon tuning of the bias voltagethe mechanical resonance frequency is tuned through the  LC  resonance, but due to the strong coupling an avoided crossingis very clearly observed. Panels ( d,e ) show the spectra corre-sponding to the orange line in ( b,c ), at  V  DC  = 242V, wherethe electronic and mechanical resonance frequencies coincide.Points are data, orange line is the fit of the model. at resonance (Ω r  ≡  Ω m  = Ω LC   = Ω), orders of magni-tude below commercial modulators optimised for decadesby the telecom industry, but also explorative microwavephotonic devices [31–33] based on electronic nonlineari-ties [34]. It is interesting to relate this performance tomore fundamental entities, namely the electromagneticfield’s quanta that constitute the signal. Indeed it is pos-sible to show that the  quantum   conversion efficiency, de-fined here as the ratio of optical sideband photons tothe rf quanta extracted from the source  V  s I/ ¯ h Ω LC  for C  em  ≫ 1, is given by (see SI) η eo  = 4( kx zpf  ) 2 Φ car Γ m .  (5)This corresponds to the squared effective Lamb-Dicke pa-rameter ( kx zpf  ) 2 = (2 π/λ ) 2 ¯ h/ (2 m Ω m ) enhanced by the  4number of photons sampling the membrane during themembrane excitations’ lifetime. We have tested that themembranes can support optical readout powers of morethan Φ car hc/λ  = 20mW without degradation of their(intrinsic) linewidth. We thus project that conversionefficiencies on the order of 50% are available. So farwe have measured a few percent in the laboratory, withfurther experiments ongoing. Note that this transducerconstitutes a phase-insensitive amplifier, and can thusreach conversion efficiencies above  one—at the expenseof added quantum noise.For the recovery of classical signals, the sensitivity andbandwidth of the interface is of greatest interest. Thesignal at the optical output of the device is the interfero-metrically measured spectral density of the optical phase ϕ  of the light reflected off the membrane, S  tot ϕϕ  = (2 k ) 2  χ eff m  2  | Gχ LC  | 2 S  VV   + S  th FF   + S  im ϕϕ ,  (6)The voltage  V  s  at the input of the resonance circuit (de-noted here as its spectral density  S  VV   ) is transducedto a phase shift via the circuit’s susceptibility  χ LC  , thecoupling  G , the effective membrane susceptibility  χ eff m and the optical wavenumber  k  (see SI). The sensitiv-ity is determined by the noise added within the inter-face. This includes in particular, the imprecision in thephase measurement ( S  im ϕϕ ), but also the random thermalmotion of the membrane induced by the Langevin force( S  th FF  ). The former depends on the performance of theemployed interferometric detector and can be quantum-limited ( S  im ϕϕ  ∼ Φ − 1car ).We demonstrate this transduction scheme by measur-ing the ambient rf radiation background [35]. This ra-diation induces a voltage  V  s  on the order of   10nV / √  Hz in the circuit, as we can determine through an electricalmeasurement on port ‘1’. Alternatively, we measure thissignal optically, as shown in Fig. 4a.In this experiment, we use a home-built interferom-eter operated at  λ  = 1064nm  and with a light powerof   0 . 8mW  returned from a thicker membrane ( d  ∼ 1 µ m ,  m  = 140ng ,  Γ m / 2 π  = 20Hz ). Optical quan-tum (shot) noise limits the phase sensitivity to   S  im ϕϕ  =20nrad / Hz , corresponding to membrane displacementsof   1 . 7fm / √  Hz . This can be translated to a voltage sensi-tivity limit by division with the total input-output trans-fer function  χ tot ≡ 2 kχ eff m  Gχ LC   of the transducer. Valuesas low as  50pV / √  Hz  are achieved within the resonantbandwidth of this proof-of-principle transducer (yellowcurve in Fig. 4b). With improved detection (probingpower  20mW , unity visibility), this number reduces to 10pV / √  Hz . Evidently, yet more sensitive optomechan-ical transduction could readily be achieved with knownsystems [17, 36]. The data in Fig. 4 corresponds to a mea-surement with about  46dB  dynamic range, and we stillexpect a significant margin towards the onset of mechan-ical nonlinearities, which were observed to occur only asthe displacement amplitudes are approaching nm-levels, FIG. 4:  Voltage sensitivity and noise.  ( a ) Optical mea-surent of ambient rf radiation (red) with a signal-to-noise ratioof nearly 50dB. The sharp peak at 771 kHz is due to cali-bration with a known phase modulation. Thick lines showmodels for the total signal (   S  tot ϕϕ , blue), with contributionsfrom the rf radiation ( √  S  VV    , violet), the optical quantumphase noise (   S  im ϕϕ , yellow), and membrane thermal noise(   S  th FF  , green). ( b ) Data and models as in (a), but dividedby the interface’s response function | χ tot | , and thus referencedto the voltage  V  s  induced in the antenna. corresponding to an expected dynamic range of   80dB  forbroadband signals.The noise added by the membrane thermal agitationis more critical in nature, as it can usually only be re-duced using cryogenic cooling. Remarkably, however,this noise is strongly suppressed in this setting. If re-cast into an equivalent input voltage noise  S  mem VV   (Ω) = S  th FF  (Ω) / | Gχ LC  (Ω) | 2 , this contribution is found as lowas S  mem VV   (Ω LC  ) = 2 k B T  m C em R,  (7)at resonance, where  T  m  is the temperature of the  mem-brane  . Thus the noise added by the membrane, as seen bythe optical readout, has a temperature of only  T  m / C em .For the data of Fig. 4, we have obtained a cooperativityof   C em  = 6800  with a bias voltage of   V  dc  = 21V . Thecorresponding noise temperature of   40mK  should there-fore allow to recover signals down to a level of   5pV / √  Hz over a bandwidth of   ∼ Γ LC   (green curve in Fig. 4b).  5For comparison, the ultralow-noise operational ampli-fier arrangement, which we use to measure on port ‘1’has a nominal voltage noise of 4nV / √  Hz and negligiblecurrent noise; in practice, with a gain of 1000, we achievean input noise (at port ‘1’) of    S  oa VV   = 17nV / √  Hz.Referenced again to the voltage  V  s  driving the circuit,this translates to a voltage sensitivity of    S  oa VV  /Q LC   =130pV / √  Hz, to be compared with the current shot-noiselimited 50pV / √  Hz of our device and its 5pV / √  Hz mem-brane noise. This superior performance of our transducerremains unchanged also off resonance (Ω  = Ω LC  ) as bothmethods benefit equally from the  LC  -circuit’s resonantenhancement of the signal voltage  V  s  (cf. Fig. 1b).Further improved sensitivity is readily available, notleast due to the still considerable margin for increasingthe cooperativity: the pull-in instability of the mem-brane is estimated to occur only at  C em  ≈  20 , 000, and S  mem VV   (Ω R ) ≈ 2 . 9pV / √  Hz in the present system (see SI).Within the stability regime, stronger coupling, and cor-respondingly higher cooperativities, can be achieved byreducing the offset capacitance  C  0 .The ultimate noise floor of our opto-mechanical trans-ducer is well below the room temperature Johnson noisefrom the circuit’s  R  = 20Ω. Our ultralow-noise trans-ducer/amplifier can therefore be of particular relevancein applications where this noise is suppressed. For ex-ample, for direct electronic (quantum) signal transduc-tion, the resonance circuit must be overloaded with acold transmission line which carries the signal of interest.In radio astronomy [35], highly efficient antennas look-ing at the cold sky have noise temperatures in the GHzrange significantly below room temperature. The usuallyrequired cryogenically cooled pre-amplifiers might be re-placed by our transducer. Finally, in nuclear-magneticresonance experiments including imaging, cooled pickupcircuits can deliver a significant sensitivity improvement,yet this approach is challenging current amplifier technol-ogy [37]. For applications with the centre radio-frequencyin the GHz band strong electro-mechanical coupling tothe membrane in the MHz range can be achieved by us-ing an oscillating coupling voltage [4, 5] instead of the dcvoltage used in the present work. Methods Summary The capacitor is fabricated by standard cleanroommicrofabrication techniques. Electrodes made of gold(thickness of 200nm) are deposited on a glass substrateand structured by ion-beam etching. Each segment is400  µ m long, with 60  µ m gaps between the segments.Pillars of a certain height (600 nm, 1 µ m) are placed todetermine the membrane-electrode distance. The capac-itor is connected in parallel to a ferrite inductor with a Q  ≈  500 (around 700 kHz) and  L  ≈  635 µ H. The in-ductor is wound with Litz wires to ensure high  Q -factor.A variable trimming capacitor is used to tune the reso-nance frequency of the LC circuit. The capacitance of themembrane-electrode system is measured to be roughly0 . 5pF.The mechanical resonator consists of a 50 nm thickAluminum layer on top of a high-stress stoichometricSiN layer with a thickness of 100 nm and 180 nm fordifferent samples. The Al layer is deposited on top of the whole wafer after the membranes have been released.Photolithography and chemical etching are subsequentlyused to remove the metal from the anchoring regions andfrom a circle in the middle of the membrane. The metallayer on SiN causes roughly a 10 percent decrease in theeigenfrequency of the fundamental mode.Optical interferometry is carried out via a commer-cial Doppler vibrometer (MSA-500 Polytec) and by ahome made Michelson interferometer (for the data setin Fig.4). The vibrometer uses optical heterodyne de-tection of the light returned from the membrane to re-cover the displacement spectrum of the membrane’s sur-face. The membrane-electrode distance is determinedusing the white light interferometry functionality of thesame device. Interference fringes are collected from thepartly reflective membrane layer and the electrode layerunderneath, which is then used to determine their rela-tive distance.The home made Michelson interferometer consists of two optical paths, namely the beam sent on the mem-brane and on a piezo-controlled mirror in the refer-ence arm. The beams are re-combined and the relativephase measured with a high-bandwidth (0-75MHz) In-GaAs balanced-homodyne receiver. Shot noise limitedmeasurement with an overall quantum efficiency of 25%(losses, visibility, detector efficiency) is achieved. Theslow signals from the two DC monitoring outputs of thedetector are used to generate the differential error signal,which is then fed to the piezo for locking the interferome-ter to the midpoint (maximum slope) of the interferencepattern. The rf output of the detector is high-pass fil-tered and fed to a spectrum analyzer in order to recordthe vibrations of the membrane. Absolute calibration of the mechanical amplitude is carried out via a calibrationpeak generated by driving the piezo with a known voltageat a frequency close to the mechanical peak. The gener-ated signal power is then converted to displacement byreferring to the visibility equation and full-fringe voltagemeasurement determined by a slow piezo scan. Acknowledgments This work was supported by the DARPA projectQUASAR, the European Union Seventh Framework Pro-gram through SIQS (grant no. 600645), the ERC grantsQIOS (grant no. 306576) and INTERFACE (grant no.291038). We would like to thank Jörg Helge Müller forvaluable discussions, Andy Barg and Andreas Næsby forassistance with the interferometer, and Louise Jørgensenfor cleanroom support.
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