Parametric symmetry breaking in a nonlinear resonator
A. Leuch, L. Papariello, O. Zilberberg, C. L. Degen, R. Chitra, A. Eichler
PParametric symmetry breaking in a nonlinear resonator
Anina Leuch, ∗ Luca Papariello, ∗ Oded Zilberberg, Christian L. Degen, R. Chitra, and Alexander Eichler Institute for Solid State Physics, ETH Zurich, 8093 Zurich, Switzerland Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Dated: September 1, 2016)Much of the physical world around us can be described in terms of harmonic oscillators in thermo-dynamic equilibrium. At the same time, the far from equilibrium behavior of oscillators is importantin many aspects of modern physics. Here, we investigate a resonating system subject to a funda-mental interplay between intrinsic nonlinearities and a combination of several driving forces. Wehave constructed a controllable and robust realization of such a system using a macroscopic dou-bly clamped string. We experimentally observe a hitherto unseen double hysteresis in both theamplitude and the phase of the resonator’s response function and present a theoretical model thatis in excellent agreement with the experiment. Our work provides a thorough understanding ofthe double-hysteretic response through a symmetry breaking of parametric phase states that elu-cidates the selection criteria governing transitions between stable solutions. Our study motivatesapplications ranging from ultrasensitive force detection to low-energy computing memory units.
PACS numbers:
Parametric excitation of resonators plays an importantrole in many areas of science and technology. In its best-known form, parametric excitation describes the modu-lation of a resonator’s natural frequency at twice the nat-ural frequency itself [1–4]. In this case, energy is pumpedinto or out of the resonator depending on the phase ofthe modulation relative to the oscillation. This ubiqui-tous feature finds applications in a wide range of fieldsincluding signal amplification and noise squeezing [5–13]with contemporary proposals also including topologicalchiral amplifiers [14], generation of quantum entangle-ment [15, 16], as well as mechanical logic operations withthe so-called parametron [17–20].The last decade has seen remarkable progress in thefabrication and control of nanomechanical resonatorswhich serve as an ideal platform for harnessing paramet-ric excitations [21–23]. As the resonators scale down,they attain unprecedented sensitivity towards minutemasses, forces and magnetic moments [24–26]. At thesame time, they enter a regime where nonlinearities be-come a defining characteristic that offers new function-ality for parametrical detectors [22, 23, 27–29]. Indeed,for sufficiently strong parametric driving, the effectivedamping of the linear resonator becomes negative and theoscillation amplitude is stabilized by nonlinearities [30].The negative effective damping regime of the paramet-ric resonator is particularly interesting because it fea-tures two stable oscillation solutions [4]. These solu-tions, which we term ‘parametric phase states’ for therest of this paper, are a result of the double periodic-ity of the parametric excitation. They are degenerate inamplitude, but phase shifted by π , and they are fasci-nating because they allow for the study of broken time-translation symmetry and activated interstate switch-ing in both classical and quantum systems [31–33]. Re-cently, it was shown that an external force field can liftthe amplitude degeneracy between the parametric phase ww V (2 w ) para V ( w ) drive V ( w ) meas x FIG. 1: Experimental realization of a parametric resonatorbased on a doubly clamped steel string (0 .
23 mm × .
23 mm × .
36 m). Direct driving at frequency ω and parametric ex-citation at frequency 2 ω rely on AC currents through coilsinduced by voltages V drive and V para , respectively [34]. Thestring position is read out from the voltage V meas induced ina pickup coil. states [29]. This degeneracy lifting becomes pronouncedin the presence of nonlinear damping and leads to a ro-bust double hysteresis in the frequency-swept responseof the resonator, which can be used to measure smallnear-resonant forces [29].In this work, we report the first experimental demon-stration of the double-hysteretic response and show thatit is intimately linked to symmetry breaking betweenparametric phase states. As a demonstrator, we use amacroscopic mechanical resonator that is similar to state-of-the-art nanomechanical resonators in terms of nonlin-ear characteristics while offering easy tuning and a signal-to-noise ratio that is rarely attained in nanomechanicaldevices. We find a complex interplay between drivingforces and nonlinearities that leads to multistability inamplitude and phase as a function of driving frequency.In parallel, we present a theoretical model that accu-rately describes our measurements and lends insight intothe governing mechanisms. As an outlook, we describeapplications that will profit directly from our study. a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug (e) IIIIIIIV w u (a)
324 325 326 -2 -1 V ( m V ) m ea s w /2 p (Hz)326.0 326.2 326.4050100 V ( m V ) m ea s w /2 p (Hz) (b) (c) V ( m V ) m ea s III III IV w /2 p (Hz)326.0 326.2 326.40 p /2 q (r ad ) m ea s (d) -p /2 w /2 p (Hz)326.0 326.2 326.4 III III IV v (f) (g)
FIG. 2: Device response to the various drives. (a) Linear response with weak external drive amplitudes V drive = 3 .
15 mV,7 .
33 mV, 17 . . . V para = 0. The background increase is due to direct electricalcoupling between the drive and detection coils. (b) Response to parametric excitation with V drive = 0 and V para = 0 . − . .
05 V. Curves are vertically offset by 10 mV for better visibility and instability boundaries are traced by gray dashedlines. Inset shows onset of instability for V para = 0 . ≡ V th . Black dots denote experimental data and all theoretical fits (redsolid lines) use Q = 1800, α = 2 . × m − s − and η = 6 . × m − s − . (c)-(d) First experimental demonstration ofdouble-hysteretic response. (c) mean displacement and (d) oscillation phase as a function of ω for both upward sweep (blackline-dots) and downward sweep (red line-dots). Four domains (I-IV) of the response appear. Here, V drive = 0 . V para = 0 . φ = − ◦ . Theory curves are gray dashed lines, cf. Eqs. (2) and (3). (e) For the fitted parameters, representative calculatedstability maps of the system in the four domains at ω = 0 . ω in I, ω = 1 . ω in II, ω = 1 . ω in III and ω = 1 . ω in IV. Stable solutions (dark red lines) and unstable solutions (bright yellow lines), as well as the bifurcations (grey spheres)as a function of ω are also shown. In (f) and (g), the corresponding solutions for purely external ( λ = 0 , F (cid:54) = 0) or parametricdrives ( λ (cid:54) = 0 , F = 0) are shown. The range of u = r cos( θ ) and v = r sin( θ ) axes corresponds to ± .
65 mm for (e) and (g), and ± .
065 mm for (f).
Our experimental setup consists of a doubly clampedsteel string, see Fig. 1. The string acts as an Euler-Bernoulli beam in the high tension limit [30]. Paramet-ric excitation is realized by modulation of the positionof one clamping point to change the tension inside thestring. The motion of the string at angular frequency ω is transduced into a voltage and read out via a lock-inamplifier [34]. The lowest energy mode of the device sat-isfies the well-known equation of motion for a nonlinear,parametrically excited resonator [30]:¨ x + ω [1 − λ cos (2 ωt )] x + Γ ˙ x + αx + ηx ˙ x = F M cos ( ωt + φ ) , (1)where x is the displacement of the resonator and dotsmark differentiations with respect to time t . The mod-ulation amplitude λ controls the parametric excitation and Γ = ω /Q is the linear damping coefficient with Q the mechanical quality factor. The nonlinearities α and η denote the conservative (Duffing-type) and dissipativenonlinearities, respectively. F is the amplitude of anapplied external force, M is the effective mass of the res-onator, which here is equal to half the total mass, and φ isa phase difference between applied force and parametricexcitation [34].We use relatively weak external driving to characterizethe linear behavior of the device. Figure 2(a) shows theresponse of the lowest mechanical mode to driving volt-ages V drive from 3 .
15 to 215 mV. Optical calibration al-lows us to translate measured voltage amplitudes, V meas ,into root mean square displacement, r , with a conversionfactor of 3 . × − m/ V meas [34]. We estimate the massto be M = 6 . × − kg from the geometry of the stringand the density of steel, and fit all response curves with ω / π ∼
325 Hz and Q = 1800. The response curveshave a purely electrical offset which grows in proportionto V drive . For small displacements relative to the stringdiameter, r < d = 2 . × − m, the peak response isproportional to V drive . This allows us to extract a linearrelationship between driving voltage and applied force as F = 4 × − N/ V drive .To access the nonlinear regime of large displacementamplitudes ( r > d ), we parametrically excite the device.In the absence of an external driving force, we measurelarge and stable vibrations for values of the paramet-ric excitation voltage V para beyond a threshold of 0 . λ th =2 /Q [30], we can calibrate the modulation amplitude as λ = λ th · V para /V th ≈ V para /
540 V. Additionally, we ob-tain the nonlinear parameters α = 2 . × m − s − and η = 1 . × m − s − by fitting all response curveswith the well-known solution of the homogeneous caseof Eq. (1), with F = 0 [30]. Our fitting relies on thefact that the nonlinear response maps the edges of theso-called “Arnold’s tongue”, i.e., the instability bound-aries of the linear parametric resonator, see dashed linesin Fig. 2(b).A striking interplay unfolds when parametric excita-tion and external driving act simultaneously. In Fig.2(c), the measured displacement amplitude for an upwardfrequency sweep exhibits a single jump (at the boundarybetween domains III and IV), akin to the jump expectedin standard externally driven Duffing resonators in theabsence of parametric excitation. However, for down-ward frequency sweeps, a double hysteresis appears andthe response displays two consecutive jumps (at the III-II and II-I boundaries, respectively). While the jumps(III-IV) and (III-II) describe the typical hysteresis for ex-ternally driven Duffing resonators, the second jump (II-I) is a novel feature that stems from an interplay withparametric excitation and has not been seen before inan experiment. The same hysteretic responses are moreprominent in the measured oscillation phase θ meas , seeFig. 2(d).For a detailed investigation of these features seen in theamplitude and phase measurements, we solve (1) usingthe approach discussed in Ref. [29]. Rewriting Eq. (1) astwo coupled first order differential equations, we use theaveraging method to obtain equations of motion for theslow-flow displacement amplitude, r , and phase, θ , of themotion ˙ r = − αrω (cid:0)
4Γ + ηr (cid:1) + 2 αkλr sin 2 θ + 4 F α sin ( θ − φ )8 ωk √ αM , (2) ˙ θ = ω ω (cid:20) αr k + 1 − ω ω − λ θ − F kr cos ( θ − φ ) (cid:21) , (3)where k = M ω . The steady state response is obtainedby setting ˙ r = ˙ θ ≡ ω [29]. Using the experimentally extracted values forthe resonator parameters, we find that the model results,which are shown in Figs. 2(c) and 2(d), are in excellentagreement with the experiment and allow an unambigu-ous interpretation of the measured phenomena.In Fig. 2(e), we plot the calculated ω -dependent stabil-ity diagram of the system. It shows the basins of attrac-tion of the system for domains I-IV along with the evolu-tion of stable attractors and unstable saddle points. Onecan see clearly how, as a function of ω , the total numberof stationary solutions increases (decreases) due to gener-ation (annihilation) of pairs of stable-unstable solutionsat bifurcation points (see grey spheres in figure). Corre-spondingly, in each domain, we have a different numberof solutions, i.e., a single solution in I, three in II, fiveand three in III, and one in IV. Following the evolutionof the stable solutions with increasing or decreasing ω reveals the origin of the second hysteretic jump in bothamplitude and phase.It is instructive to compare the stability diagram inFig. 2(e) to its two limiting cases, namely the sys-tem in the presence of purely external driving or purelyparametric excitation. The corresponding stability di-agrams are shown in Figs. 2(f) and 2(g), respectively.Intuitively, we can construct the full stability diagramfrom the purely parametric case by regarding the exter-nal drive as a perturbation. As a consequence of thisperturbation, the parametric phase states are no longersymmetric and the trivial solutions ( r = 0) seen in Fig.2(g) are shifted toward finite amplitudes. Importantly,an opposite phase is imprinted on the stationary solu-tion by the external drive in regions I and IV. As a com-bined result of these two effects, for opposing directionsof frequency sweeps, a different parametric phase state ischosen and the double hysteresis is seen.The central observation of the theoretical analysisabove is that the amplitude degeneracy and phase sym-metry of parametric phase states are broken by the ex-ternal driving force. In order to verify this claim, we ex-perimentally probe the stability diagram of the resonatorin domain II, see Fig. 3(a). We prepare the resonator ata fixed frequency and with low amplitude using a smallexternal drive. The phase of the external drive, or equiv-alently the phase difference φ , determines the startingposition of the resonator on the inner circle ( ∼ φ determines which parametric phase state (a) (b) -2 -3 V ( m V ) m ea s V para V drive p 0p/ FIG. 3: Role of relative phase φ between drives. Pulseschematics at the bottom illustrate the order of driv-ing/excitation voltages as a function of time for the two exper-iments. (a) We prepare the resonator at low amplitude r with V drive = 0 . V para = 0 . φ . Radial scale is logarithmic. Inset on up-per left schematically indicates frequency. (b) Reversing theorder of (a), we prepare the resonator with V para = 0 . V drive = 0 . φ . Here, scale is linear. is chosen, the double hysteresis is visible only for an ap-propriate range of φ . This is systematically explored inthe supplemental material [34].To show the one-to-one correspondence between thepure and perturbed parametric phase states, we preparethe resonator in one of the pure phase states with onlyparametric excitation, see Fig. 3(b). When the exter-nal drive is added, the resonator solution shifts awayfrom - and settles on a ring around - the center of theplot. Again, the final position on the ring depends on φ .Apart from the fact that the perturbed attractors canbe mapped onto the original parametric phase states,this experiment also demonstrates that the perturbedphase states are stable for all values of φ , even if thesystem would preferably select the opposite phase state[as shown in Fig. 3(a)].In addition to ultrasensitive force detection as pro-posed in Ref. [29], the double hysteresis opens up newpossibilities to control the parametron, a digital storageelement that utilizes the parametric phase states to en-code bits [17]. Optical and mechanical manifestationsof coupled parametrons, where each phase state doubletplays the role of an artificial spin, are being developedto emulate and solve Ising Hamiltonians [35, 36]. Inconventional computing, the parametron is a candidatefor low-energy consumption memory [18]. As we visual-ize in Fig. 4 for the example of a nanomechanical res-onator, the double hysteresis allows switching betweenphase states in a controlled manner without changing w drive time II IIIVI w state ‘0’state ‘ p ’ (a) (b) V para V DC V drive V para V drive V DC FIG. 4: Proposed new parametron control sequence enabledby double hysteresis. (a) Schematic parametric NEMS devicewith electrodes for drive tones and control voltage. V para , V drive and V DC refer to a parametric drive tone applied at2 ω drive , an external drive tone at ω drive and a DC controlvoltage, respectively. (b) In the proposed control sequence,the driving frequency ω drive and the relative phase φ remainfixed, whereas the resonance frequency of the resonator istuned by V DC . Whenever ω drive coincides with domain II, theresonator switches to one of the parametric phase states ‘0’or ‘ π ’ (shaded regions). Which state is chosen depends on thedomain in which the resonator was prepared previously. amplitude, frequency or phase of the external drive tone(applied at ω drive ) and the parametric drive tone (appliedat 2 ω drive ). Parametric state switching is performed bychanging the resonance frequency of the nanomechanicalresonator with a small DC voltage. The parametron re-sides in one of the two phase states whenever domain IIof the resonator overlaps with ω drive . Which phase stateis selected depends entirely on whether the resonator waspreviously prepared in domain I or IV. The parametroncan thus be fully controlled by changing a DC voltage.For the device used in Ref. [18], a DC voltage of ∼ . µ V range.The analysis that we present here is valid for anynonlinear resonator subject to a combination of para-metric excitation and external driving. Such resonatorsare actively studied in many modern fields of physics,with examples ranging from levitating nanoparticles [37],coupled photonic microcavities [38], nanomechanical res-onators and optomechanics to quantum electrodynam-ics [39]. Future directions include generalization of ourresults to quantum systems and coupled resonators in thepresence of noise.The authors gratefully acknowledge technical supportand know-how from P. M¨arki, C. Keck, U. Grob andT. Ihn. The construction of the setup was done in col-laboration with the engineering office (M. Baer) and themechanical workshop at the Department of Physics atETH Zurich. This work has been supported by the ERCthrough Starting Grant 309301, and by the Swiss Na-tional Science Foundation. ∗ Authors contributed equally.[1] M. Faraday, Philosophical transactions of the Royal So-ciety of London , 299 (1831).[2] ´E. Mathieu, Journal de math´ematiques pures et ap-pliqu´ees , 137 (1868).[3] L. Rayleigh, The London, Edinburgh, and Dublin Philo-sophical Magazine and Journal of Science , 145 (1887).[4] L. Landau and E. Lifshitz, Mechanics , Butterworth-Heinemann (1976).[5] P. Penfield and R. P. Rafuse,