Linear and nonlinear photoinduced deformations of cantilevers

Glassy and elastomeric nematic networks with dye molecules present can be very responsive to illumination, huge reversible strains being possible. If absorption is appreciable, strain decreases with depth into a cantilever, leading to bend that is the basis of micro-opto-mechanical systems (MOMS). Bend actually occurs even when Beer's law suggests a tiny penetration of light into a heavily dye-doped system. We model the nonlinear opto-elastic processes behind this effect. In the regime of cantilever thickness giving optimal bending for a given incident light intensity, there are three neutral surfaces. In practice such nonlinear absorptive effects are very important since heavily doped systems are commonly used.     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Controlled generation of intrinsic localized modes in microelectromechanical cantilever arrays

We propose a scheme to induce intrinsic localized modes (ILMs) at an arbitrary site in microelectromechanical cantilever arrays. The idea is to locate the particular cantilever beam in the array that one wishes to drive to an oscillating state with significantly higher amplitude than the average and then apply small adjustments to the electrical signal that drives the whole array system. Our scheme is thus a global closed-loop control strategy. We argue that the dynamical mechanism on which our global driving scheme relies is spatiotemporal chaos and we develop a detailed analysis based on the standard averaging method in nonlinear dynamics to understand the working of our control scheme. We also develop a Markov model to characterize the transient time required for inducing ILMs.     

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.     

Controlling chaos in dynamic-mode atomic force microscope

We successfully demonstrated the first experimental stabilization of irregular and non-periodic cantilever oscillation in the amplitude modulation atomic force microscopy using the time-delayed feedback control. A perturbation to cantilever excitation force stabilized an unstable periodic orbit associated with nonlinear cantilever dynamics. Instead of the typical piezoelectric excitation, the magnetic excitation was used for directly applying control force to the cantilever. The control force also suppressed the cantilever's occasional bouncing motions that caused artifacts on a surface image.     

Matter wave soliton collisions in the quasi one-dimensional potential

We consider soliton solutions of a two-dimensional nonlinear system with the self-focusing nonlinearity and a quasi 1D confining potential, taking harmonic potential as an example. We investigate a single soliton in detail and find criterion for possible collapse. This information is then used to investigate the dynamics of the two soliton collision. In this dynamics we identify three regimes according to the relation between nonlinear interaction and the excitation energy: elastic collision, excitation and collapse regime. We show that surprisingly accurate predictions can be obtained from variational analysis.     

Bose-Einstein condensate coupled to a nanomechanical resonator on an atom chip

We theoretically study the coupling of Bose-Einstein condensed atoms to the mechanical oscillations of a nanoscale cantilever with a magnetic tip. This is an experimentally viable hybrid quantum system which allows one to explore the interface of quantum optics and condensed matter physics. We propose an experiment where easily detectable atomic spin flips are induced by the cantilever motion. This can be used to probe thermal oscillations of the cantilever with the atoms. At low cantilever temperatures, as realized in recent experiments, the backaction of the atoms onto the cantilever is significant and the system represents a mechanical analog of cavity quantum electrodynamics. With high but realistic cantilever quality factors, the strong coupling regime can be reached, either with single atoms or collectively with Bose-Einstein condensates. We discuss an implementation on an atom chip.     

An analytical estimate of the period for the delayed logistic application and the Lotka-Volterra system

We first introduce a simple and new method for the quantitative analysis of some nonlinear oscillating systems. It is shown that if the dynamics of the system reduces to piecewise exponential growth and exponential damping phases, then the amplitude and period of the motion can be computed with accuracy in the nonlinear regime without invoking linear stability arguments or perturbative expansions. This method is then successfully applied to the delayed logistic application and to the Lotka-Volterra prey-predator model. For both of these systems, we provide an accurate analytical expression for the period of the oscillations in the nonlinear regime. Received: 27 April 1998 / Received in final form: 25 June 1998 / Accepted: 29 June 1998     

Spin Squeezing and Fixed-Point Bifurcation

We study spin squeezing and classical bifurcation in a nonlinear bipartite system. We show that the spin squeezing can be associated with a fixed-point bifurcation in the classical dynamics, namely, it acts as an indicator of the classical bifurcation. For the ground state of a system with coupled giant spins, we find that the spin squeezing achieves its minimum value near the bifurcation point. We also study the dynamics of the spin squeezing, for an initial state corresponding to one of the fixed point, we find that in the stable regime, the spin squeezing exhibits periodic oscillation and always persists except at some fixed times, while in the unstable regime, the periodic oscillation phenomenon disappears and the spin squeezing survives for a short time. Finally, we show that the mean spin squeezing, which is defined to be averaged over time, attains its minimum value near the bifurcation point.     

Spin Squeezing and Fixed-Point Bifurcation

We study spin squeezing and classical bifurcation in a nonlinear bipartite system. We show that the spin squeezing can be associated with a fixed-point bifurcation in the classical dynamics, namely, it acts as an indicator of the classical bifurcation. For the ground state of a system with coupled giant spins, we find that the spin squeezing achieves its minimum value near the bifurcation point. We also study the dynamics of the spin squeezing, for an initial state corresponding to one of the fixed point, we find that in the stable regime, the spin squeezing exhibits periodic oscillation and always persists except at some fixed times, while in the unstable regime, the periodic oscillation phenomenon disappears and the spin squeezing survives for a short time. Finally, we show that the mean spin squeezing, which is defined to be averaged over time, attains its minimum value near the bifurcation point.     

Dynamics of microcantilever integrated with geometric nonlinearity for stable and broadband nonlinear atomic force microscopy

We explore the use of a nonlinear cantilever system integrating geometric nonlinearity for AFM imaging, in contrast from the traditional linear cantilever system. The intrinsically nonlinear AFM cantilever system exhibits broadband resonance over a bandwidth several times of its linear resonant frequency and possesses an intrinsic stability that virtually eliminates the instability induced by the tip–sample interactions involved in a linear AFM system, thus the artifact of image contrast reversal. The ability to realize broadband operation may extend the application of AFM to spectral analysis of tip–sample interactions across a broad frequency range at the nanoscale.     

Stabilization and utilization of nonlinear phenomena based on bifurcation control for slow dynamics

Mechanical systems may experience undesirable and unexpected behavior and instability due to the effects of nonlinearity of the systems. Many kinds of control methods to decrease or eliminate the effects have been studied. In particular, bifurcation control to stabilize or utilize nonlinear phenomena is currently an active topic in the field of nonlinear dynamics. This article presents some types of bifurcation control methods with the aim of realizing vibration control and motion control for mechanical systems. It is also indicated through every control method that slowly varying components in the dynamics play important roles for the control and the utilizations of nonlinear phenomena. In the first part, we deal with stabilization control methods for nonlinear resonance which is the 1/3-order subharmonic resonance in a nonlinear spring-mass-damper system and the self-excited oscillation (hunting motion) in a railway vehicle wheelset. The second part deals with positive utilizations of nonlinear phenomena by the generation and the modification of bifurcation phenomena. We propose the amplitude control method of the cantilever probe of an atomic force microscope (AFM) by increasing the nonlinearity in the system. Also, the motion control of a two link underactuated manipulator with a free link and an active link is considered by actuating the bifurcations produced under high-frequency excitation. This article is a discussion on the bifurcation control methods presented by the author and co-researchers by focusing on the actuation of the slowly varying components included in the original dynamics.     

Nonlinear dynamics of modulation instability in distributed resonators under external harmonic driving

We study the regimes of complex field dynamics upon modulation instability in distributed nonlinear resonators under external harmonic driving. Two regimes are considered: the regime of a nonlinear ring cavity, described by nonlinear Schrödinger equation (NLS) with a delayed boundary condition, and the regime of a one-dimensional Fabri-Perot cavity, described by a system of coupled NLS for the forward and backward waves. Theoretical stability analysis of stationary forced oscillations is carried out. The results of numerical simulation of transition to chaos with increasing input intensity are presented.     

Observation of an amplitude collapse and revival of chirped coherent phonons in bismuth

We have studied the A(1g) coherent phonons in bismuth generated by high fluence ultrashort laser pulses. We observed that the nonlinear regime, where the phonons' oscillation parameters depend on fluence, consists of subregimes with distinct dynamics. Just after entering the nonlinear regime, the phonons become chirped. Increasing the fluence further leads to the emergence of a collapse and revival, which next turns into multiple collapses and revivals. This is explained by the dynamics of a wave packet in an anharmonic potential, where the packet periodically breaks up and reconstitutes in its original form, giving convincing evidence that the phonons are in a quantum state, with no classical analog.     

Strong-coupling regime of the nonlinear landau-zener problem for photo- and magnetoassociation of cold atoms

We discuss the strong-coupling regime of the nonlinear Landau-Zener problem occurring at coherent photo- and magneto-association of ultracold atoms. We apply a variational approach to an exact third-order nonlinear differential equation for the molecular state probability and construct an accurate approximation describing the time dynamics of the coupled atom-molecule system. The resultant solution improves the accuracy of the previous approximation [22]. The obtained results reveal a remarkable observation that in the strong-coupling limit, the resonance crossing is mostly governed by the nonlinearity, while the coherent atom-molecule oscillations occurring soon after crossing the resonance are principally of a linear nature. This observation is supposedly general for all nonlinear quantum systems having the same generic quadratic nonlinearity, due to the basic attributes of the resonance crossing processes in such systems. The constructed approximation turns out to have a larger applicability range than it was initially expected, covering the whole moderate-coupling regime for which the proposed solution accurately describes ail the main characteristics of the system evolution except the amplitude of the coherent atom-molecule oscillation, which is rather overestimated.     

Opto-Mechanical Effects in Superradiant Light Scattering by Bose-Einstein Condensate in an Optical Cavity

We investigate the effects of a movable mirror (cantilever) of an optical cavity on the superradiant light scattering from a Bose-Einstein condensate (BEC) in an optical lattice. We show that the mirror motion has a dynamic dispersive effect on the cavity-pump detuning. Varying the intensity of the pump beam, one can switch between the pure superradiant regime and the Bragg scattering regime. The mechanical frequency of the mirror strongly influences the time interval between two Bragg peaks. We find that when the system is in the resolved side band regime for mirror cooling, the superradiant scattering is enhanced due to coherent energy transfer from the mechanical mirror mode to the cavity field mode.     

Opto-Mechanical Effects in Superradiant Light Scattering by Bose-Einstein Condensate in an Optical Cavity

We investigate the effects of a movable mirror (cantilever) of an optical cavity on the superradiant light scattering from a Bose-Einstein condensate (BEC) in an optical lattice. We show that the mirror motion has a dynamic dispersive effect on the cavity-pump detuning. Varying the intensity of the pump beam, one can switch between the pure superradiant regime and the Bragg scattering regime. The mechanical frequency of the mirror strongly influences the time interval between two Bragg peaks. We find that when the system is in the resolved side band regime for mirror cooling, the superradiant scattering is enhanced due to coherent energy transfer from the mechanical mirror mode to the cavity field mode.     

Collective atomic motion in an optical lattice formed inside a high finesse cavity

We report on collective nonlinear dynamics in an optical lattice formed inside a high finesse ring cavity in a so far unexplored regime, where the light shift per photon times the number of trapped atoms exceeds the cavity resonance linewidth. We observe bistability and self-induced squeezing oscillations resulting from the retroaction of the atoms upon the optical potential wells. We can well understand most of our observations within a simplified model assuming adiabaticity of the atomic motion. Nonadiabatic aspects of the atomic motion are reproduced by solving the complete system of coupled nonlinear equations of motion.     

Essay: Preheating and Turbulence: Echoes of a Not So Quiet Universe

We study the nonlinear decay of the inflaton which causes the reheating of the Universe in the transition from the inflationary phase to the radiation dominated phase, resulting in the creation of almost all matter constituting the present Universe. Our treatment allows us to follow the full dynamics of the system in a long time regime, and to describe not only the parametric resonance processes with nonlinear restructuring but also to characterize a final turbulent state in the dynamics by which the energy is nonlinearly transferred to all scales of the system with a consequent thermalization of the created matter.     

Nonlinear resonances of a single-wall carbon nanotube cantilever

The dynamics of an electrostatically actuated carbon nanotube (CNT) cantilever are discussed by theoretical and numerical approaches. Electrostatic and intermolecular forces between the single-walled CNT and a graphene electrode are considered. The CNT cantilever is analyzed by the Euler–Bernoulli beam theory, including its geometric and inertial nonlinearities, and a one-mode projection based on the Galerkin approximation and numerical integration. Static pull-in and pull-out behaviors are adequately represented by an asymmetric two-well potential with the total potential energy consisting of the CNT elastic energy, electrostatic energy, and the Lennard-Jones potential energy. Nonlinear dynamics of the cantilever are simulated under DC and AC voltage excitations and examined in the frequency and time domains. Under AC-only excitation, a superharmonic resonance of order 2 occurs near half of the primary frequency. Under both DC and AC loads, the cantilever exhibits linear and nonlinear primary and secondary resonances depending on the strength of the excitation voltages. In addition, the cantilever has dynamic instabilities such as periodic or chaotic tapping motions, with a variation of excitation frequency at the resonance branches. High electrostatic excitation leads to complex nonlinear responses such as softening, multiple stability changes at saddle nodes, or period-doubling bifurcation points in the primary and secondary resonance branches.