Anomalous Thermostat and Intraband Discrete Breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation for the anharmonic coordinates. For zero temperature, we prove that the support of the Fourier transform of the memory kernel and of the time averaged velocity-velocity correlations functions of the anharmonic system can not overlap. As a consequence, the asymptotic solutions can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency $\Omega$ we find that the energy $E_T$ transferred to the harmonic system up to time $T$ is proportional to $T^{\alpha}$. If $\Omega$ equals one of the phonon frequencies $\omega_\nu$, it is $\alpha=2$. We prove that there is a full measure set such that for $\Omega$ in this set it is $\alpha=0$, i.e. there is no energy dissipation. Under certain conditions there exists a zero measure set such that for $\Omega \in this set the dissipation rate is nonzero and may be subdissipative$(0 \leq \alpha < 1)$or superdissipative$(1 <\alpha \leq 2)$. Consequently, the harmonic bath does act as an anomalous thermostat. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency$\Omega$exist for all$\Omega$in a Cantor set$\mathcal{C}(k)$of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing in the memory kernel. For$\Omega\in\mathcal{C}(k)$the small denominators do not lead to divergencies such that this kernel is a smooth and bounded function in$t$.Comment: Physica D in prin

Discrete Breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm

A Quantum Non-demolition measurement of Fock states of mesoscopic mechanical oscillators

We investigate a scheme that makes a quantum non-demolition measurement of the excitation level of a mesoscopic mechanical oscillator by utilizing the anharmonic coupling between two elastic beam bending modes. The non-linear coupling between the two modes shifts the resonant frequency of the readout oscillator proportionate to the excitation of the system oscillator. This frequency shift may be detected as a phase shift of the readout oscillation when driven on resonance. We show that in an appropriate regime this measurement approaches a quantum non-demolition measurement of the phonon number of the system oscillator. As phonon energies in micromechanical oscillators become comparable to or greater than the thermal energy, the individual phonon dynamics within each mode can be resolved. As a result it should be possible to monitor jumps between Fock states caused by the coupling of the system to the thermal reservoirs.Comment: revised, 21 pages, 9 figure

ANALYSIS OF THE NONLINEAR VIBRATIONS OF ELECTROSTATICALLY ACTUATED MICRO-CANTILEVERS IN HARMONIC DETECTION OF RESONANCE (HDR)

Micro- and nano-cantilevers have the potential to revolutionize physical, chemical, and biological sensing; however, an accurate and scalable detection method is required. In this work, a fully electrical actuation and detection method is presented, known as the Harmonic Detection of Resonance (HDR). In HDR, harmonic components of the current are measured to determine the cantilever\u27s resonance frequency. These harmonics exist as a result of nonlinearities in the system, principally in the electrostatic actuation force. In order to better understand this rich harmonic structure, a theoretical investigation of the micro-cantilever is undertaken. Both a lumped parameter model and a more accurate continuum model are used to derive the governing nonlinear modal equations of motion (EOM) of the cantilever. Various approximate solution methods applicable to nonlinear equations are then discussed including numerical integration, perturbation, and averaging. An averaging method known as the method of harmonic balance is then used to obtain steady state solutions to the micro-cantilever EOM. Low-order closed-form harmonic balance solutions are derived which explain many of the important features of the HDR results, such as the presence of parasitic capacitance in the first harmonic and super-harmonic resonance peaks in higher harmonics. Finally, higher-order computer generated harmonic balance solutions are presented which show good agreement with the experimental HDR results, validating both the modeling and the solution methods used