APPLIED PHYSICS LETTERS 90, 043101 ͑2007͒

Detection of coherent acoustic oscillations in a quantum electromechanical resonator Florian W. Beil Center for NanoScience and Sektion Physik, Ludwigs-Maximilians-Universität, Geschwister-Scholl-Platz 1, 80539 München, Germany ͒ Robert H. Blicka Department of Electrical and Computer Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, Wisconsin 53706 Achim Wixforth Lehrstuhl für Experimentalphysik I, Universität Augsburg, Universitätsstraße 1, 86159 Augsburg, Germany Werner Wegscheider and Dieter Schuh Institut für Angewandte und Experimentelle Physik, Universität Regensburg, 93040 Regensburg, Germany Max Bichler Walter Schottky Institut, Am Coloumbwall 3, 85748 Garching, Germany ͑Received 20 September 2006; accepted 18 December 2006; published online 22 January 2007͒ Coherent control of occupation numbers in quantum mechanical multilevel systems is widely studied driven by its application in lasers and its prospects for quantum computational elements. Here the authors present a nanoelectromechanical resonator equivalent to the coherent control of a quantum mechanical two level system. The distinct eigenmodes of a nanomechanical beam resonator represent the two levels whose amplitude mode occupation numbers are controlled by a frequency matched acoustic excitation, mediated by a pulsed surface acoustic wave. They show that similar to quantum mechanical systems it is possible to transfer occupation numbers from one mode to another by matched acoustic pulses. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2432954͔ A straightforward method to control occupation numbers quency between two eigenmodes of a nanomechanical beam in quantum mechanical two level systems is to apply short resonator ͑see Fig. 1͒.5 In full equivalence to atoms pumped pulses of light, which are frequency matched to the transition by light, the resonator represents the multilevel system, frequency between the two levels.1 The transfer of occupa- whereas the phonons excited by the acoustic wave corre- tion number between two levels, usually termed as Rabi os- spond to the photons. The amplitude of the resonator beam’s cillations, relies on a system’s state which can be described first eigenmode shows similar behavior as observed for oc- as a superposition of distinct modes, attributed eigenstates, cupation numbers in the quantum mechanical equivalent, being coupled by an external perturbation of the system. This namely, Rabi-like oscillations versus pump power and a pumping of atoms into an excited state results in an occupa- Rabi-like splitting.6 tion inversion, which is the precondition for lasing operation. To demonstrate the mechanical equivalence to pumped Consequently, a mechanical analog can be realized once atoms, the suspended, gold covered nanobeam is placed in eigenstates are identified and appropriately addressed. For the line of fire of the interdigitated transducer ͑IDT͒ gener- quantum electromechanical ͑QEM͒ systems, such as nano- ating the SAW. The comblike IDT structure allows genera- mechanical resonators, the different eigenstates correspond tion of SAWs by the inverse piezoelectric effect, which con- to different vibrational modes. Any displacement of the reso- verts an applied radio frequency ͑rf͒ signal to the electrodes nator can be described as a superposition of eigenmodes, into mechanical stress, as the electric potential drops be- whereas each eigenmode j contributes with an amplitude or, tween each finger pair. If the frequency fsaw of the signal ␭ in analogy to the quantum mechanical counterpart, mode oc- meets the resonance condition fsaw=vsaw/ , a coherent cupation number Aj to the overall motion. Usually, it is not acoustic sound wave is generated. The IDT, however, pro- possible to directly control the mode occupation numbers Aj, vides a certain bandwidth, being determined by the number since thermal noise or nonlinear coupling determines the of finger pairs, hence allowing for a moderate frequency tun- mode distribution. In the macroscopic extreme classical cou- ing, as employed in Fig. 3, for example, Ref. 5. The fre- −1 pling of modes can occur for suspension bridges due to quency is defined by the sound velocity vsaw=2865 m s pulsed wind excitation, as the Takoma bridge disaster has and the lithographically defined SAW wavelength ␭ ͑corre- 2 shown or due to pedestrians crossing the Millenium Bridge sponding to the pitch between the SAW electrodes͒. As the 3 in London. length of the beam L is matched to ␭/2, the two clamping In the experiment described here a surface acoustic wave points move in counterphase and the beam is periodically 4 ͑SAW͒ of Rayleigh type, properly described as a nano- stressed by this motion. quake, on GaAs is frequency matched to the transition fre- The setup is placed in a strong magnetic field, which allows magnetomotive excitation and detection of the beam’s ͒ 7 a Author to whom correspondence should be addressed; electronic mail: off-plane eigenmode. Driving an alternating current at fre- [email protected] quency f with power Pres along the conduction top metallic

0003-6951/2007/90͑4͒/043101/3/$23.0090, 043101-1 © 2007 American Institute of Physics Downloaded 01 Feb 2007 to 129.187.254.47. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 043101-2 Beil et al. Appl. Phys. Lett. 90, 043101 ͑2007͒

FIG. 1. ͑Color online͒ Top and lower right show setup of the experiment. The suspended beam is placed in the line of fire of a transducer which ͑ ͒ generates a surface acoustic wave SAW of frequency fsaw. The SAW dis- places the anchoring points of the resonator beam in counterphase and pe- riodically stresses the beam. The inset right of the center shows an actual SEM micrograph of the resonator beam and the clamping points. An in- plane magnetic field B allows detection of the beam’s off-plane eigenmode by impedance spectroscopy. Lower left: schematic representation of the SAW induced pumping from the first to the third eigenmode of the beam ␻ ␻ ⍀ 3 − 1 = saw.

layer causes a Lorentz force, which when in resonance ex- cites the beam’s mechanical modes with an amplitude A.In Fig. 2͑a͒ we traced the power dependence of this amplitude with A=Ares. The induced motion changes the beam’s acous- ͑ ͒ FIG. 2. a Dependence of the resonator oscillation amplitude A=Ares on the toelectrical impedance Zbeam, which we measure as an effec- applied magneto impedance probe power Pres. Above −56 dBm the nonlin- tive amplitude of the oscillations A=A in Fig. 2͑b͒. In turn ear coupling to higher modes leads to a deviation from the linear increase. osc ͑ ͒ ͑ ͒ it exhibits a mechanical resonance in the reflected power. b Upper part: amplitudes A=Aosc of acoustic Rabi-like oscillations ARO ϳ͑ ͒ for different probe powers Pres. The AROs start to vanish for probe powers Measuring the scattering parameter S11 1−Aosc versus above the nonlinear limit of −56 dBm. Lower part: visualization of the ARO ϳ͑ ͒ driving frequency f reveals the fundamental mechanical in the reflectance S11 1−Aosc at the beam vs acoustic pulse power Psaw. ϳ resonance, shown in Fig. 2. In this plot an increased modal occupation A A1 corresponds to darker AϳA areas, whereas the lighter regions correspond to low A1, due to acoustic For large amplitudes res, nonlinear effects come coupling to a higher eigenmode A . into play and distribute energy among the beam’s eigen- 3 ͑ ͒ modes. This is shown in Fig. 2 a , where we plot Ares against the probe power Pres applied at the beam. Above P= Another characteristic feature which the quantum elec- −56 dBm nonlinear effects kick in and higher eigenmodes tromechanical resonator reveals is the characteristic depen- ͑ ϳ ͒ are excited Ares A1 +A3 . The necessary adjustment of the dence of the ARO on pump frequency fsaw as expected from ⍀ ␲ quantum mechanical systems. In Fig. 3͑a͒ the measured am- SAW frequency saw=2 fsaw to the transition frequency of plitude AϳA is plotted against a broad range of SAW fre- the beam’s first eigenmode to its third harmonic ftrans= f3 1 ϳ␻ ␻ ⍀ quencies. This shows that the observed mode coupling only − f1 3 − 1 = saw was calculated by finite element simula- tions, achieving sufficient accuracy for the IDT’s bandwidth occurs for a specific pulse center frequency Fsaw= ftrans. ͑Fig. 1͒. Comparing the measured curve to the Fourier transform of A f For pumping the transition from the first to the third the applied SAW signal FT at trans allows us to extract the transition frequency of the first to the third eigenmode to be eigenmode, short acoustic pulses were applied. The minimal 302.9 MHz, which lies in the active frequency range of the pulse length is determined by the pitch of the IDT and is of IDT. Here, we are able to cover a broader frequency range, the order of 100 ns. In Fig. 2͑b͒ A=A ϳA exhibits oscil- osc 1 since we operate the IDTs in pulsed mode. Hence, we find lations against increased SAW pulse amplitude Asaw. This evenly spaced harmonics in the range of 280–340 MHz. The effect is best visualized when coding the reflected signal shape of A vs f around the transition frequency for two ͓ ͑ ͔͒ saw from the beam in a grayscale plot Fig. 2 c . These oscilla- different SAW powers shows the form of two separated reso- ϳ͑ ͒ tions of S11 1−Aosc vs Asaw are equivalent to oscillations nances ͓Fig. 3͑b͔͒. This again is expected for the coupling of of occupation numbers in a two level system usually plotted two states leading to Rabi-like splitting of the energy levels. versus pump power; hence we term these oscillations as To model acoustic pumping of mode transitions we acoustical Rabi-like oscillations ͑AROs͒. Increasing the adopt the theory developed for quantum mechanical systems magnetoimpedance probe power Pres, the AROs disappear to the acoustic case. The amplitude of the beam resonator’s when increasing the power above the limit of nonlinear re- mechanical amplitude A in the linear regime is described as a ͓ ͑ ͔͒ sponse at −56 dBm Fig. 2 b . Pres corresponds to the power superposition of eigenmodes Aj, whereas each mode j con- ␺ ␺ of the network analyzer applied to the sample. tributes to it with specific wave function shapes 1 and 3. Downloaded 01 Feb 2007 to 129.187.254.47. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 043101-3 Beil et al. Appl. Phys. Lett. 90, 043101 ͑2007͒

͑ ͒ FIG. 3. a ARO vs SAW frequency fsaw and calculated component of the ͑ ͒ FIG. 4. Simulated ͑top͒ and measured ͑bottom͒ AROs in amplitude A.To applied acoustic signal’s Fourier transform AFT at f = ftrans. From this ftrans is evaluated to f =303.2 MHz. ͑b͒ Measured amplitude A for f close to model a contribution from the third eigenmode to the reflected signal to the trans saw ϳ first one, a superposition of A A1 +0.5A3 was calculated for different SAW ftrans for two different SAW pulse powers. The acoustic frequency splitting decreases for lower P . pulse amplitudes. The simulation shows vanishing ARO for increased occu- saw ͑ ͒ pation of A3 compared to A1, as observed in Fig. 2 a . The SAW induced periodic stress acting on the beam is in- cluded in the Euler-Bernoulli beam equations by an addi- shown in Fig. 4. For the different traces the parameters A1 8,9 tional term, which will make the Aj become time depen- and A are varied in combination as A +0.5A , ensuring that 10 3 1 3 dent. Using the orthogonality of eigenmodes in the rotating the fundamental mode at A and the higher harmonic at A 5 1 3 wave approximation, which assumes that contributions from are excited. The calculations resemble the experimentally ob- eigenmodes with frequencies very different from f1 + ftrans served ARO in Fig. 4 for the range of applied SAW ampli- can be neglected, we obtain the following coupled differen- tudes. tial equation for the acoustically coupled eigenmodes A1 and In summary, we demonstrated that in analogy to the con- A3: trol of occupation numbers in a quantum mechanical two -level system, the mode occupation numbers in quantum elec ץ 2ץ A1 A1 +2i␻ =−⍀2 A , tromechanical resonators can be controlled by pulsed acous- t 13 3ץ t2 1ץ tic excitation. Hence, it is possible to coherently control the mode occupation numbers in a nanomechanical resonator, Aץ 2Aץ 3 ␻ 3 ⍀2 which can be exploited for achieving occupation inversion +2i 3 =− 13A1. t necessary for a mechanical equivalent to lasers, termed theץ t2ץ ␻ ␲ ⍀ “phaser.” Here, j =2 f j, and 13 is the acoustic Rabi-like frequency, which is determined by the mass per unit length ␮ of the This work was supported by the Deutsche Forschungs- beam, the mismatch between the SAW frequency and the gemeinschaft ͑Bl-487/3͒ and the Air Force RSO ͑F49620-03- ⌬ ͒ transition frequency = ftrans− fsaw. 1-0420 . L 2 F 1 ץ ⍀ = ͵ ␺ ␺ dx sawei⌬t. A. Zrenner, E. Beham, S. Stufler, F. Findeis, M. Bichler, and G. Abstreiter, .␮ Nature ͑London͒ 418, 612 ͑2002͒ 3 2 ץ 1 13 0 x 2K. Y. Billak and R. H. Scanlan, Am. J. Phys. 59,118͑1991͒. 3 This integral depends on the mode shapes ␺ and ␺ of the S. H. Strogatz, D. M. Abrams, A. McRobie, and B. Eckhardt, Nature 1 3 ͑ ͒ ͑ ͒ two coupled modes integrated over the length L of the beam London 483,43 2005 . 4L. D. Landau and E. M. Lifhitz, Theory of Elasticity ͑Butterworth- and the force Fsaw the SAW exerts on the beam. The force Heinemann, Oxford, 1999͒, Vol. 1, p. 97. can be calculated from simple beam theory with the assump- 5A. Wixforth, J. Scriba, M. Wassermeier, J. P. Kotthaus, G. Weimann, and tion of elliptic motion of the suspension points with ampli- W. Schlapp, Phys. Rev. B 40, 7874 ͑1989͒. 6I. I. Rabi, N. F. Ramsey, and J. Schwinger, Rev. Mod. Phys. 26, 167 tude Asaw. ͑1954͒. The two coupled modes will periodically exchange en- 7R. H. Blick, A. Erbe, L. Pescini, A. Kraus, D. V. Scheible, F. W. Beil, E. ergy, whereas the acoustic Rabi frequency determines the M. Höhberger, A. Hoerner, J. Kirschbaum, H. Lorenz, and J. P. Kotthaus, period of this exchange. The amplitude of the SAW can be J. Phys.: Condens. Matter 14, R905 ͑2002͒. estimated by the acoustic power injected by the IDT, leading 8K. Magnus and K. Popp, Schwingungen ͑Teubner, Stuttgart, 1997͒,Vol.1, to typical frequencies of ⍀ in the megahertz regime. This p. 15. 13 9K. F. Graff, Wave Motion in Anelastic Solids ͑Ohio State University Press, corresponds to the observed Rabi-like splitting in Fig. 3, Columbus, 1975͒, Vol. 1, p. 42. ⍀ 10 which allows direct evaluation of 13. Solving the coupled C. Cohen-Tannoudji, Quantum Mechanics ͑Wiley, New York, 1996͒,Vol. differential equation ͓Eq. ͑1͔͒ leads to the theoretical curves 2, p. 137. Downloaded 01 Feb 2007 to 129.187.254.47. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp