Casimir Effect on the Pull-in Parameters of Nanometer Switches

Casimir effect on the critical pull-in gap and pull-in voltage of nanoelectromechanical switches is studied. An approximate analytical expression of the critical pull-in gap with Casimir force is presented by the perturbation theory. The corresponding pull-in parameters are computed numerically, from which one can notice the nonlinear effect of Casimir force on the pull-in parameters. The detachment length has been presented, which increases with increasing thickness of the beam.

Rheological effect on thermocapillary flow of a liquid film jet painted on a moving boundary

In the present paper, a liquid (or melt) film of relatively high temperature ejected from a vessel and painted on the-moving solid film is analyzed by using the second-order fluid model of the non-Newtonian fluid. The thermocapillary flow driven by the temperature gradient on the free surface of a Newtonian liquid film was discussed before. The effect of rheological fluid on thermocapillary flow is considered in the present paper. The analysis is based on the approximations of lubrication theory and perturbation theory. The equation of liquid height and the process of thermal hydrodynamics of the non-Newtonian liquid film are obtained, and the case of weak effect of the rheological fluid is solved in detail.

Effect of surface tension on the mode selection of vertically excited surface waves in a circular cylindrical vessel

Singular perturbation theory of two-time-scale expansions was developed in inviscid fluids to investigate patternforming, structure of the single surface standing wave, and its evolution with time in a circular cylindrical vessel subject to a vertical oscillation. A nonlinear slowly varying complex amplitude equation, which involves a cubic nonlinear term, an external excitation and the influence of surface tension, was derived from the potential flow equation. Surface tension was introduced by the boundary condition of the free surface in an ideal and incompressible fluid. The results show that when forced frequency is low, the effect of surface tension on the mode selection of surface waves is not important. However, when the forced frequency is high, the surface tension cannot be neglected. This manifests that the function of surface tension is to cause the free surface to return to its equilibrium configuration. In addition, the effect of surface tension seems to make the theoretical results much closer to experimental results.

Surface waves in a vertically excited circular cylindrical container

The nonlinear free surface amplitude equation, which has been derived from the inviscid fluid by solving the potential equation of water waves with a singular perturbation theory in a vertically oscillating rigid circular cylinder, is investigated successively in the fourth-order Runge-Kutta approach with an equivalent time-step. Computational results include the evolution of the amplitude with time, the characteristics of phase plane determined by the real and imaginary parts of the amplitude, the single-mode selection rules of the surface waves in different forced frequencies, contours of free surface displacement and corresponding three-dimensional evolution of surface waves, etc. In addition, the comparison of the surface wave modes is made between theoretical calculations and experimental measurements, and the results are reasonable although there are some differences in the forced frequency.

Diffusion dominated process for the crystal growth of a binary alloy

The pure diffusion process has been often used to study the crystal growth of a binary alloy in the microgravity environment. In the present paper, a geometric parameter, the ratio of the maximum deviation distance of curved solidification and melting interfaces from the plane to the radius of the crystal rod, was adopted as a small parameter, and the analytical solution was obtained based on the perturbation theory. The radial segregation of a diffusion dominated process was obtained for cases of arbitrary Peclet number in a region of finite extension with both a curved solidification interface and a curved melting interface. Two types of boundary conditions at the melting interface were analyzed. Some special cases such as infinite extension in the longitudinal direction and special range of Peclet number were reduced from the general solution and discussed in detail.

Non-axisymmetric magnetostatic equilibrium with strong twisted component of magnetic field

The perturbation theory is applied further to the discussion of the equilibrium properties of a sunspot-like magnetic field with a strong twisted component. The basic state reduces to the usual one discussed extensively for the axisymmetric magnetostatic equilibrium with twisted component of magnetic field, and the perturbed state is described by two coupled equations. As the magnetic force-line is twisted, there is a magnetic tension in the azimuthal direction. In this case, the perturbed total pressure is no longer independent of the azimuthal variable θ, and the magnetic field in the dark penumbal fibril may be either stronger or weaker relatively.

Some modified bifurcation problems with application to imperfection sensitivity in buckling

The branching theory of solutions of certain nonlinear elliptic partial differential equations is developed, when the nonlinear term is perturbed from unforced to forced. We find families of branching points and the associated nonisolated solutions which emanate from a bifurcation point of the unforced problem. Nontrivial solution branches are constructed which contain the nonisolated solutions, and the branching is exhibited. An iteration procedure is used to establish the existence of these solutions, and a formal perturbation theory is shown to give asymptotically valid results. The stability of the solutions is examined and certain solution branches are shown to consist of minimal positive solutions. Other solution branches which do not contain branching points are also found in a neighborhood of the bifurcation point.

The qualitative features of branching points and their associated nonisolated solutions are used to obtain useful information about buckling of columns and arches. Global stability characteristics for the buckled equilibrium states of imperfect columns and arches are discussed. Asymptotic expansions for the imperfection sensitive buckling load of a column on a nonlinearly elastic foundation are found and rigorously justified.

Polaronic correction to the first excited electronic energy level in an anisotropic semiconductor quantum dot

Within the framework of second-order Rayleigh-Schrodinger perturbation theory, the polaronic correction to the first excited state energy of an electron in an quantum dot with anisotropic parabolic confinements is presented. Compared with isotropic confinements, anisotropic confinements will make the degeneracy of the excited states to be totally or partly lifted. On the basis of a three-dimensional Frohlich's Hamiltonian with anisotropic confinements, the first excited state properties in two-dimensional quantum dots as well as quantum wells and wires can also be easily obtained by taking special limits. Calculations show that the first excited polaronic effect can be considerable in small quantum dots.

Charged pion contribution to the anomalous magnetic moment of the muon

The model dependence inherent in hadronic calculations is one of the dominant sources of uncertainty in the theoretical prediction of the anomalous magnetic moment of the muon. In this thesis, we focus on the charged pion contribution and turn a critical eye on the models employed in the few previous calculations of $a_\mu^{\pi^+\pi^-}$. Chiral perturbation theory provides a check on these models at low energies, and we therefore calculate the charged pion contribution to light-by-light (LBL) scattering to $\mathcal{O}(p^6)$. We show that the dominant corrections to the leading order (LO) result come from two low energy constants which show up in the form factors for the $\gamma\pi\pi$ and $\gamma\gamma\pi\pi$ vertices. Comparison with the existing models reveal a potentially significant omission - none include the pion polarizability corrections associated with the $\gamma\gamma\pi\pi$ vertex. We next consider alternative models where the pion polarizability is produced through exchange of the $a_1$ axial vector meson. These have poor UV behavior, however, making them unsuited for the $a_\mu^{\pi^+\pi^-}$ calculation. We turn to a simpler form factor modeling approach, generating two distinct models which reproduce the pion polarizability corrections at low energies, have the correct QCD scaling at high energies, and generate finite contributions to $a_\mu^{\pi^+\pi^-}$. With these two models, we calculate the charged pion contribution to the anomalous magnetic moment of the muon, finding values larger than those previously reported: $a_\mu^\mathrm{I} = -1.779(4)\times10^{-10}\,,\,a_\mu^\mathrm{II} = -4.892(3)\times10^{-10}$.

Topics in gravitational-wave science : macroscopic quantum mechanics and black hole physics

The theories of relativity and quantum mechanics, the two most important physics discoveries of the 20th century, not only revolutionized our understanding of the nature of space-time and the way matter exists and interacts, but also became the building blocks of what we currently know as modern physics. My thesis studies both subjects in great depths --- this intersection takes place in gravitational-wave physics.

Gravitational waves are "ripples of space-time", long predicted by general relativity. Although indirect evidence of gravitational waves has been discovered from observations of binary pulsars, direct detection of these waves is still actively being pursued. An international array of laser interferometer gravitational-wave detectors has been constructed in the past decade, and a first generation of these detectors has taken several years of data without a discovery. At this moment, these detectors are being upgraded into second-generation configurations, which will have ten times better sensitivity. Kilogram-scale test masses of these detectors, highly isolated from the environment, are probed continuously by photons. The sensitivity of such a quantum measurement can often be limited by the Heisenberg Uncertainty Principle, and during such a measurement, the test masses can be viewed as evolving through a sequence of nearly pure quantum states.

The first part of this thesis (Chapter 2) concerns how to minimize the adverse effect of thermal fluctuations on the sensitivity of advanced gravitational detectors, thereby making them closer to being quantum-limited. My colleagues and I present a detailed analysis of coating thermal noise in advanced gravitational-wave detectors, which is the dominant noise source of Advanced LIGO in the middle of the detection frequency band. We identified the two elastic loss angles, clarified the different components of the coating Brownian noise, and obtained their cross spectral densities.

The second part of this thesis (Chapters 3-7) concerns formulating experimental concepts and analyzing experimental results that demonstrate the quantum mechanical behavior of macroscopic objects - as well as developing theoretical tools for analyzing quantum measurement processes. In Chapter 3, we study the open quantum dynamics of optomechanical experiments in which a single photon strongly influences the quantum state of a mechanical object. We also explain how to engineer the mechanical oscillator's quantum state by modifying the single photon's wave function.

In Chapters 4-5, we build theoretical tools for analyzing the so-called "non-Markovian" quantum measurement processes. Chapter 4 establishes a mathematical formalism that describes the evolution of a quantum system (the plant), which is coupled to a non-Markovian bath (i.e., one with a memory) while at the same time being under continuous quantum measurement (by the probe field). This aims at providing a general framework for analyzing a large class of non-Markovian measurement processes. Chapter 5 develops a way of characterizing the non-Markovianity of a bath (i.e.,whether and to what extent the bath remembers information about the plant) by perturbing the plant and watching for changes in the its subsequent evolution. Chapter 6 re-analyzes a recent measurement of a mechanical oscillator's zero-point fluctuations, revealing nontrivial correlation between the measurement device's sensing noise and the quantum rack-action noise.

Chapter 7 describes a model in which gravity is classical and matter motions are quantized, elaborating how the quantum motions of matter are affected by the fact that gravity is classical. It offers an experimentally plausible way to test this model (hence the nature of gravity) by measuring the center-of-mass motion of a macroscopic object.

The most promising gravitational waves for direct detection are those emitted from highly energetic astrophysical processes, sometimes involving black holes - a type of object predicted by general relativity whose properties depend highly on the strong-field regime of the theory. Although black holes have been inferred to exist at centers of galaxies and in certain so-called X-ray binary objects, detecting gravitational waves emitted by systems containing black holes will offer a much more direct way of observing black holes, providing unprecedented details of space-time geometry in the black-holes' strong-field region.

The third part of this thesis (Chapters 8-11) studies black-hole physics in connection with gravitational-wave detection.

Chapter 8 applies black hole perturbation theory to model the dynamics of a light compact object orbiting around a massive central Schwarzschild black hole. In this chapter, we present a Hamiltonian formalism in which the low-mass object and the metric perturbations of the background spacetime are jointly evolved. Chapter 9 uses WKB techniques to analyze oscillation modes (quasi-normal modes or QNMs) of spinning black holes. We obtain analytical approximations to the spectrum of the weakly-damped QNMs, with relative error O(1/L^2), and connect these frequencies to geometrical features of spherical photon orbits in Kerr spacetime. Chapter 11 focuses mainly on near-extremal Kerr black holes, we discuss a bifurcation in their QNM spectra for certain ranges of (l,m) (the angular quantum numbers) as a/M → 1. With tools prepared in Chapter 9 and 10, in Chapter 11 we obtain an analytical approximate for the scalar Green function in Kerr spacetime.

Singularities and horizons in the collisions of gravitational waves

This thesis presents a study of the dynamical, nonlinear interaction of colliding gravitational waves, as described by classical general relativity. It is focused mainly on two fundamental questions: First, what is the general structure of the singularities and Killing-Cauchy horizons produced in the collisions of exactly plane-symmetric gravitational waves? Second, under what conditions will the collisions of almost-plane gravitational waves (waves with large but finite transverse sizes) produce singularities?

In the work on the collisions of exactly-plane waves, it is shown that Killing horizons in any plane-symmetric spacetime are unstable against small plane-symmetric perturbations. It is thus concluded that the Killing-Cauchy horizons produced by the collisions of some exactly plane gravitational waves are nongeneric, and that generic initial data for the colliding plane waves always produce "pure" spacetime singularities without such horizons. This conclusion is later proved rigorously (using the full nonlinear theory rather than perturbation theory), in connection with an analysis of the asymptotic singularity structure of a general colliding plane-wave spacetime. This analysis also proves that asymptotically the singularities created by colliding plane waves are of inhomogeneous-Kasner type; the asymptotic Kasner axes and exponents of these singularities in general depend on the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

In the work on collisions of almost-plane gravitational waves, first some general properties of single almost-plane gravitational-wave spacetimes are explored. It is shown that, by contrast with an exact plane wave, an almost-plane gravitational wave cannot have a propagation direction that is Killing; i.e., it must diffract and disperse as it propagates. It is also shown that an almost-plane wave cannot be precisely sandwiched between two null wavefronts; i.e., it must leave behind tails in the spacetime region through which it passes. Next, the occurrence of spacetime singularities in the collisions of almost-plane waves is investigated. It is proved that if two colliding, almost-plane gravitational waves are initially exactly plane-symmetric across a central region of sufficiently large but finite transverse dimensions, then their collision produces a spacetime singularity with the same local structure as in the exact-plane-wave collision. Finally, it is shown that a singularity still forms when the central regions are only approximately plane-symmetric initially. Stated more precisely, it is proved that if the colliding almost-plane waves are initially sufficiently close to being exactly plane-symmetric across a bounded central region of sufficiently large transverse dimensions, then their collision necessarily produces spacetime singularities. In this case, nothing is now known about the local and global structures of the singularities.

Phase conjugation via four-wave mixing in a resonant medium

This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process.

In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed.

In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile.

Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given.

Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10^-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.

Studies of phonon anharmonicity in solids

Today our understanding of the vibrational thermodynamics of materials at low temperatures is emerging nicely, based on the harmonic model in which phonons are independent. At high temperatures, however, this understanding must accommodate how phonons interact with other phonons or with other excitations. We shall see that the phonon-phonon interactions give rise to interesting coupling problems, and essentially modify the equilibrium and non-equilibrium properties of materials, e.g., thermodynamic stability, heat capacity, optical properties and thermal transport of materials. Despite its great importance, to date the anharmonic lattice dynamics is poorly understood and most studies on lattice dynamics still rely on the harmonic or quasiharmonic models. There have been very few studies on the pure phonon anharmonicity and phonon-phonon interactions. The work presented in this thesis is devoted to the development of experimental and computational methods on this subject.

Modern inelastic scattering techniques with neutrons or photons are ideal for sorting out the anharmonic contribution. Analysis of the experimental data can generate vibrational spectra of the materials, i.e., their phonon densities of states or phonon dispersion relations. We obtained high quality data from laser Raman spectrometer, Fourier transform infrared spectrometer and inelastic neutron spectrometer. With accurate phonon spectra data, we obtained the energy shifts and lifetime broadenings of the interacting phonons, and the vibrational entropies of different materials. The understanding of them then relies on the development of the fundamental theories and the computational methods.

We developed an efficient post-processor for analyzing the anharmonic vibrations from the molecular dynamics (MD) calculations. Currently, most first principles methods are not capable of dealing with strong anharmonicity, because the interactions of phonons are ignored at finite temperatures. Our method adopts the Fourier transformed velocity autocorrelation method to handle the big data of time-dependent atomic velocities from MD calculations, and efficiently reconstructs the phonon DOS and phonon dispersion relations. Our calculations can reproduce the phonon frequency shifts and lifetime broadenings very well at various temperatures.

To understand non-harmonic interactions in a microscopic way, we have developed a numerical fitting method to analyze the decay channels of phonon-phonon interactions. Based on the quantum perturbation theory of many-body interactions, this method is used to calculate the three-phonon and four-phonon kinematics subject to the conservation of energy and momentum, taking into account the weight of phonon couplings. We can assess the strengths of phonon-phonon interactions of different channels and anharmonic orders with the calculated two-phonon DOS. This method, with high computational efficiency, is a promising direction to advance our understandings of non-harmonic lattice dynamics and thermal transport properties.

These experimental techniques and theoretical methods have been successfully performed in the study of anharmonic behaviors of metal oxides, including rutile and cuprite stuctures, and will be discussed in detail in Chapters 4 to 6. For example, for rutile titanium dioxide (TiO₂), we found that the anomalous anharmonic behavior of the B_1g mode can be explained by the volume effects on quasiharmonic force constants, and by the explicit cubic and quartic anharmonicity. For rutile tin dioxide (SnO₂), the broadening of the B_2g mode with temperature showed an unusual concave downwards curvature. This curvature was caused by a change with temperature in the number of down-conversion decay channels, originating with the wide band gap in the phonon dispersions. For silver oxide (Ag₂O), strong anharmonic effects were found for both phonons and for the negative thermal expansion.

I. Current commutators in quantum electrodynamics. II. Electromagnetic mass differences of the pseudoscalar meson and baryon octets

This thesis is in two parts. In the first section, the operator structure of the singular terms in the equal-time commutator of space and time components of the electromagnetic current is investigated in perturbation theory by establishing a connection with Feynman diagrams. It is made very plausible that the singular term is a c number. Some remarks are made about the same problem in the electrodynamics of a spinless particle.

In the second part, an SU(3) symmetric multi-channel calculation of the electromagnetic mass differences in the pseudoscalar meson and baryon octets is carried out with an attempt to include some of the physics of the crossed (pair annihilation) channel along the lines of the recent work by Ball and Zachariasen. The importance of the tensor meson Regge trajectories is emphasized. The agreement with experiment is poor for the isospin one mass differences, but excellent for those with isospin two.

Dynamical models for the Earth's geoid

The Earth's largest geoid anomalies occur at the lowest spherical harmonic degrees, or longest wavelengths, and are primarily the result of mantle convection. Thermal density contrasts due to convection are partially compensated by boundary deformations due to viscous flow whose effects must be included in order to obtain a dynamically consistent model for the geoid. These deformations occur rapidly with respect to the timescale for convection, and we have analytically calculated geoid response kernels for steady-state, viscous, incompressible, self-gravitating, layered Earth models which include the deformation of boundaries due to internal loads. Both the sign and magnitude of geoid anomalies depend strongly upon the viscosity structure of the mantle as well as the possible presence of chemical layering.

Correlations of various global geophysical data sets with the observed geoid can be used to construct theoretical geoid models which constrain the dynamics of mantle convection. Surface features such as topography and plate velocities are not obviously related to the low-degree geoid, with the exception of subduction zones which are characterized by geoid highs (degrees 4-9). Recent models for seismic heterogeneity in the mantle provide additional constraints, and much of the low-degree (2-3) geoid can be attributed to seismically inferred density anomalies in the lower mantle. The Earth's largest geoid highs are underlain by low density material in the lower mantle, thus requiring compensating deformations of the Earth's surface. A dynamical model for whole mantle convection with a low viscosity upper mantle can explain these observations and successfully predicts more than 80% of the observed geoid variance.

Temperature variations associated with density anomalies in the man tie cause lateral viscosity variations whose effects are not included in the analytical models. However, perturbation theory and numerical tests show that broad-scale lateral viscosity variations are much less important than radial variations; in this respect, geoid models, which depend upon steady-state surface deformations, may provide more reliable constraints on mantle structure than inferences from transient phenomena such as postglacial rebound. Stronger, smaller-scale viscosity variations associated with mantle plumes and subducting slabs may be more important. On the basis of numerical modelling of low viscosity plumes, we conclude that the global association of geoid highs (after slab effects are removed) with hotspots and, perhaps, mantle plumes, is the result of hot, upwelling material in the lower mantle; this conclusion does not depend strongly upon plume rheology. The global distribution of hotspots and the dominant, low-degree geoid highs may correspond to a dominant mode of convection stabilized by the ancient Pangean continental assemblage.