Collective and Stochastic Effects in Arrays of Submicron Oscillators

Transcription

1 DYNAMICS DAYS: Long Beach, Collective and Stochastic Effects in Arrays of Submicron Oscillators Ron Lifshitz (Tel Aviv), Jeff Rogers (HRL, Malibu), Oleg Kogan (Caltech), Yaron Bromberg (Tel Aviv), Alexander Zumdieck (Max Planck, Dresden) Support: NSF, Nato and EU, BSF, HRL

2 DYNAMICS DAYS: Long Beach, Outline Motivation: MEMS and NEMS Familiar examples of dynamics in a MEMS context Pattern formation in parametrically driven arrays Synchronization of arrays of oscillators Noise driven transitions between nonequilibrium states Conclusions

3 DYNAMICS DAYS: Long Beach, Array of µ-scale oscillators [From Buks and Roukes (2002)]

4 DYNAMICS DAYS: Long Beach, Single crystal silicon [From Craighead, Science (2000)]

5 DYNAMICS DAYS: Long Beach, MicroElectroMechanical Systems and NEMS Arrays of tiny mechanical oscillators: driven, dissipative nonequilibrium nonlinear collective noisy (potentially) quantum

6 DYNAMICS DAYS: Long Beach, MicroElectroMechanical Systems and NEMS Arrays of tiny mechanical oscillators: driven, dissipative nonequilibrium nonlinear collective noisy (potentially) quantum New laboratory for nonlinear dynamics and pattern formation Apply knowledge from nonlinear dynamics, pattern formation etc. to technologically important questions Investigate pattern formation and nonlinear dynamics in new regimes Study new aspects of old questions

7 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n

8 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n with δ n taken from distribution g(δ n )

9 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + m D nm (x n+m 2x n + x n m )

10 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 )

11 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 ) + xn 3

12 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 ) + xn 3 [ ] + η (x n+1 x n ) 2 (ẋ n+1 ẋ n ) (x n x n 1 ) 2 (ẋ n ẋ n 1 )

13 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 ) + xn 3 [ ] + η (x n+1 x n ) 2 (ẋ n+1 ẋ n ) (x n x n 1 ) 2 (ẋ n ẋ n 1 ) γ ẋ n (1 x 2 n )

14 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 ) + xn 3 [ ] + η (x n+1 x n ) 2 (ẋ n+1 ẋ n ) (x n x n 1 ) 2 (ẋ n ẋ n 1 ) + g P cos [(2 + δω P )t] x n

15 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 ) + xn 3 [ ] + η (x n+1 x n ) 2 (ẋ n+1 ẋ n ) (x n x n 1 ) 2 (ẋ n ẋ n 1 ) + g P cos [(2 + δω P )t] x n + g D cos [(1 + δω D )t]

16 DYNAMICS DAYS: Long Beach, Modelling 0 =ẍ n + x n + δ n x n + D nm (x n+m 2x n + x n m ) m γ(ẋ n+1 2ẋ n +ẋ n 1 ) + xn 3 [ ] + η (x n+1 x n ) 2 (ẋ n+1 ẋ n ) (x n x n 1 ) 2 (ẋ n ẋ n 1 ) + g P cos [(2 + δω P )t] x n + g D cos [(1 + δω D )t] + Noise

17 DYNAMICS DAYS: Long Beach, Theoretical Approach Oscillators at frequency unity + small corrections Assume dispersion, coupling, damping, driving, noise, and nonlinear terms are small. Introduce small parameter ε with ε p characterizing the size of these various terms. Then with the slow time scale T = εt [ ] x n (t) = A n (T )e it + c.c. + ɛx n (1) (t) +... derive equations for da n /dt =.

18 DYNAMICS DAYS: Long Beach, Nonlinearity in MEMS A simple nonlinear oscillator: the Duffing equation ẍ + γ ẋ + x + x 3 = g D cos(ω D t) Parameters: γ g D ω D damping drive strength drive frequency For chosen sign of x 3 term the spring gets stiffer with increasing displacement.

19 DYNAMICS DAYS: Long Beach, Frequency pulling small driving large driving Amplitude x/g D Frequency ω D

20 DYNAMICS DAYS: Long Beach, Experiment [Buks and Roukes, 2001]

21 DYNAMICS DAYS: Long Beach, [Buks and Roukes, 2001]

22 DYNAMICS DAYS: Long Beach, Parametric Drive in MEMS ẍ + γ ẋ + (1 + g P cos ω P t)x + x 3 = 0 oscillation of parameter of equation here the spring constant x = 0 remains a solution in the absence of noise parametric drive decreases effective dissipation (for one quadrature of oscillations) amplification for small drive amplitudes instability for large enough drive amplitudes strongest response for ω p = 2

23 DYNAMICS DAYS: Long Beach, MEMS Elastic Parametric Drive [Harrington and Roukes]

24 DYNAMICS DAYS: Long Beach, [Harrington and Roukes]

25 DYNAMICS DAYS: Long Beach, Amplification [Harrington and Roukes]

26 DYNAMICS DAYS: Long Beach, Parametric Instability in Arrays of Oscillators [Buks and Roukes, 2001]

27 DYNAMICS DAYS: Long Beach, Simple Intuition Frequency ω ω P ω P /2 π/a -k P Wave vector k k P π/a Above the parametric instability nonlinearity is essential to understand the oscillations. Mode Competition Pattern formation

28 DYNAMICS DAYS: Long Beach,

29 DYNAMICS DAYS: Long Beach, One Beam Theory Fixed g P Drive amplitude g P instability amplification Intensity x 2 Include nonlinear damping 2 Drive frequency ω P 2 Drive frequency ω P

30 DYNAMICS DAYS: Long Beach, Two Beam Theory Fixed Fixed 0 =ẍ n + x n + x 3 n + 2 (1 + g P cos [(2 + ε P )t])(x n+1 2x n + x n 1 ) γ(ẋ n+1 2ẋ n +ẋ n 1 ) [ ] + η (x n+1 x n ) 2 (ẋ n+1 ẋ n ) (x n x n 1 ) 2 (ẋ n ẋ n 1 ) Local Duffing (elasticity) + Electrostatic Coupling (dc and modulated) + Dissipation (currents) + Nonlinear Damping (also currents) [Lifshitz and MCC Phys. Rev. B67, (2003)]

31 DYNAMICS DAYS: Long Beach, Periodic Solutions stable unstable x s 2 x u 2 Ω Ω Intensity of symmetric mode x s 2 and antisymmetric mode x u 2 as frequency is scanned.

32 DYNAMICS DAYS: Long Beach, Periodic Solutions stable unstable x s 2 x u 2 Ω Ω The green lines correspond to a single excited mode, the remainder to coupled modes.

33 DYNAMICS DAYS: Long Beach, Hysteresis for Two Beams 5 4 Response 3 2 S 1 D 1 1 S 2 D Ω

34 DYNAMICS DAYS: Long Beach, Simulations of 67 Beams Response <x n 2 > 10 3 Sweep up Sweep down ÿ Drive Frequency ω p /ω 0

35 DYNAMICS DAYS: Long Beach, Many Beams Single oscillator Vary pump frequency Many oscillators Fixed pump frequency Drive amplitude g P instability amplification Drive amplitude g P patterns uniform state stable 2 Drive frequency ω P ω(k)=ω P Wavevector k Continuum approximation: new amplitude equation [Bromberg, MCC and Lifshitz (preprint, 2005)] A T = A + 2 A X 2 + i 2 ( 4 A 2 A ) A + A2 2 A 2 A A 4 A 3 X X

36 DYNAMICS DAYS: Long Beach, Synchronization Huygen s Clocks (1665) From: Bennett, Schatz, Rockwood, and Wiesenfeld (Proc. Roy. Soc. Lond. 2002)

37 DYNAMICS DAYS: Long Beach, Paradigm I: Synchronization occurs through dissipation acting on the phase differences Huygen s clocks (cf. Bennett, Schatz, Rockwood, and Wiesenfeld) Winfree-Kuramoto phase equation θ n = ω n m K nm sin(θ n θ n+m ) with ω n taken from distribution g(ω). Continuum limit (short range coupling) θ = ω(x) + K 2 θ + O( ( θ) 3 ) phase diffusion, not propagation (eg. no ( θ) 2 term) Aronson, Ermentrout and Kopell analysis of two coupled oscillators Matthews, Mirollo and Strogatz magnitude-phase model

38 DYNAMICS DAYS: Long Beach, Synchronization in MEMS alternative mechanism Paradigm II: Synchronization occurs by nonlinear frequency pulling and reactive coupling MEMS equation 0 =ẍ n +(1+ω n )x n ν(1 x 2 n )ẋ n ax 3 n + m D nm (x n+m 2x n + x n m ) leads to Ȧ n = i(ω n α A n 2 )A n + (1 A n 2 )A n + i m β mn (A m A n ) with a α, D β. (cf. Synchronization by Pikovsky, Rosenblum, and Kurths)

39 DYNAMICS DAYS: Long Beach, Synchronization in MEMS alternative mechanism Paradigm II: Synchronization occurs by nonlinear frequency pulling and reactive coupling MEMS equation 0 =ẍ n +(1+ω n )x n ν(1 x 2 n )ẋ n ax 3 n + m D nm (x n+m 2x n + x n m ) leads to Ȧ n = i(ω n α A n 2 )A n + (1 A n 2 )A n + i m β mn (A m A n ) with a α, D β. (cf. Synchronization by Pikovsky, Rosenblum, and Kurths) Analyze mean field version (all-to-all coupling): β mn β/n

40 DYNAMICS DAYS: Long Beach, Definitions of Synchronization 1. Order parameter = N 1 n A n = N 1 n r n e iθ n = Re i Synchronization occurs if R = 0 2. Full locking: θ n = for all the oscillators 3. Partial frequency locking 4. ω n = lim T θ n (T ) θ n (0) T and then ω n = for some O(N) subset of oscillators

41 DYNAMICS DAYS: Long Beach, Results for the mean field phase model (Kuramoto 1975) 1.0 Unlocked Partially locked Fully locked R K c K

42 DYNAMICS DAYS: Long Beach, Calculations [MCC, Zumdieck, Lifshitz, and Rogers (2004)] Linear instability of unsynchronized R = 0 state (for Lorentzian, triangular, top-hat g(ω)) Linear instability of fully locked state Simulations of amplitude-phase model for up to oscillators with all-to-all coupling

43 DYNAMICS DAYS: Long Beach, Calculations [MCC, Zumdieck, Lifshitz, and Rogers (2004)] Linear instability of unsynchronized R = 0 state (for Lorentzian, triangular, top-hat g(ω)) Linear instability of fully locked state Simulations of amplitude-phase model for up to oscillators with all-to-all coupling Results Order parameter frequency = not trivially given by g(ω) For fixed α>α min there are two values of β giving linear instability Linear instability of fully locked state may be through stationary or Hopf bifurcation No amplitude death as in Matthews et al. Complicated phase diagram with regions of coexisting states

44 DYNAMICS DAYS: Long Beach, Results for a triangular distribution g(ω) 2/w -w/2 w/2 ω Show results for w = 2

45 DYNAMICS DAYS: Long Beach, Unsynchronized state unstable α 0.5 Unsynchronized state stable β

46 DYNAMICS DAYS: Long Beach, Synchronized Fully locked stable α 0.5 Fully locked unstable Unsynchronized state stable β

47 DYNAMICS DAYS: Long Beach, Synchronized Fully locked stable α 0.5 Fully locked unstable Unsynchronized state stable β

48 DYNAMICS DAYS: Long Beach,

49 DYNAMICS DAYS: Long Beach, Synchronized Fully locked α 0.5 Partially locked Unsynchronized Unsynchronized + Synchronized β

50 DYNAMICS DAYS: Long Beach, Synchronized Fully locked α 0.5 Partially locked Unsynchronized Unsynchronized + Synchronized β

51 DYNAMICS DAYS: Long Beach,

52 DYNAMICS DAYS: Long Beach, Synchronized Fully locked α Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized β

53 DYNAMICS DAYS: Long Beach, Synchronized Fully locked α Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized β

54 DYNAMICS DAYS: Long Beach,

55 DYNAMICS DAYS: Long Beach, Synchronized Fully locked α Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized β

56 DYNAMICS DAYS: Long Beach, Synchronized Fully locked α Synchronized States Partially locked Unsynchronized Unsynchronized + Synchronized β

57 DYNAMICS DAYS: Long Beach, Noise Induced Transitions between Nonequilibrium States Amplitude Noise induced transitions Frequency

58 DYNAMICS DAYS: Long Beach, Current Status Lots of theory for single oscillators, few experimental tests No (?) results for many degree of systems Connection with non-equilibrium potential (Graham ). In weak noise limit η 0 P(x) exp ( (x)/η) Smoothness properties of (x)? Prediction of deterministic dynamics from (x)? Recent experiments

59 DYNAMICS DAYS: Long Beach, [From Aldridge and Cleland (cond-mat/ , 2004)]

60 DYNAMICS DAYS: Long Beach, Conclusions Sub-micron oscillator arrays provide a new laboratory for nonlinear and nonequilibrium physics New features: importance of noise and eventually quantum effects as dimensions shrink discreteness Motivates new directions for theoretical investigation Physics of pattern formation, synchronization etc. may be useful in technological applications