We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}

Generation of Non-classical Motional States of a Trapped Atom

Nearly a century after Einstein first predicted the existence of gravitational waves, a global network of Earth-based gravitational wave observatories1, 2, 3, 4 is seeking to directly detect this faint radiation using precision laser interferometry. Photon shot noise, due to the quantum nature of light, imposes a fundamental limit on the attometre-level sensitivity of the kilometre-scale Michelson interferometers deployed for this task. Here, we inject squeezed states to improve the performance of one of the detectors of the Laser Interferometer Gravitational-Wave Observatory (LIGO) beyond the quantum noise limit, most notably in the frequency region down to 150 Hz, critically important for several astrophysical sources, with no deterioration of performance observed at any frequency. With the injection of squeezed states, this LIGO detector demonstrated the best broadband sensitivity to gravitational waves ever achieved, with important implications for observing the gravitational-wave Universe with unprecedented sensitivity.

Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light

Theory and Application of the Z-Transform Method

In this study, problems related to the thermoelastic modeling and behavior of thin-walled beams made of functionally graded materials (FGMs) are addressed. In this context, two structural systems are considered: (i) rotating turbomachinery blades, and (ii) thin-walled beam structures spinning about their longitudinal axis. In all these cases, the structures operate in a high temperature environment. Under the spinning speed, gyroscopic forces are generated. Although conservative in nature, under their influence, as in nonconservative systems, instabilities by flutter and divergence can occur. In this context, the implications on their vibration and instability of conservative and gyroscopic forces considered in conjunction with a temperature field that yields the degradation of elastic properties are investigated. A continuously graded variation in composition of the ceramic and metal phases across the beam wall thickness in terms of a simple power law distribution is implemented. Results highlig...

Thin-Walled Beams Made of Functionally Graded Materials and Operating in a High Temperature Environment: Vibration and Stability

Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia

An analytical procedure, based on the dynamic stiffness method, is proposed for the study of rotor dynamics problems. On the grounds of the governing differential equations of a continuous beam, the dynamic stiffness matrix of the rotating Timoshenko beam is derived, including the effects of translational and rotational inertia, gyroscopic moments and shear deformation of the shaft. Concentrated disks and isotropic, elastic bearings are taken into account in the element formulation. The results obtained by the proposed method are compared with both classical closed form solutions and finite element analyses taken from the literature.

An analytical approach to the dynamics of rotating shafts

The behavior of linear rotor-bearing systems is investigated by using the exact approach of the dynamic stiffness method, which entails the use of continuous rather than lumped models. In particular, the theoretical formulation for rotor systems with anisotropic bearings is developed by utilizing the complex representation of all the involved variables. The proposed formulation eventually leads to the 8 x 8 complex dynamic stiffness matrix of the rotating Timoshenko beam; this matrix proves to be related, by a simple rule, to the 4 X 4 dynamic stiffness matrix, which describes rotor systems with isotropic bearings. The method is first applied to the critical speeds evaluation of a simple rotor system with rigid supports; for this case, the exact results of the dynamic stiffness approach are compared to the usual convergence procedure of the finite element method. Successively, the steady-state unbalance response of two rotor systems with anisotropic supports is analyzed; for these examples, the dynamic stiffness results compare favorably with the results of the finite element and the transfer matrix analysis performed by other authors.

The dynamic stiffness method for linear rotor-bearing systems

We study the nonlinear quantum Rabi model—both in the two-photon and the intensity-dependent coupling scheme—by utilizing the perturbation method in the off- and near-resonance regimes characterized by the field and the atomic transition frequencies. We consider the weak-coupling condition, the perturbation parameter being the ratio between the coupling constant and the atomic frequency. For both models we determine the first- and second-order corrections to energy eigenvalues and eigenstates, disclosing the doublet structure characterizing the energy spectrum. Then we apply our findings to both the coupling schemes in order to calculate the time evolution of a general initial state and the atomic population inversion for an initial state composed by a mixture of excited and unexcited atoms. [Phys. Rev. A 93 (2016) 043814]

Off-resonance regimes in nonlinear quantum Rabi models

It is shown that the Mathieu eigenvalues can be straightforwardly utilized to study the instability regions of an axially loaded simply supported shaft, the shaft being modeled as a continuous rotating beam. When a harmonic axial load is taken into account, the equation of motion of the system, here written according to the Lagrangian formulation of continuous systems, proves to be the Mathieu equation. It follows that the conditions for the stable or unstable motion of the shaft can be graphically investigated once the operation line corresponding to an actual rotating shaft is drawn in the Mathieu map.

Dynamic instability of axially loaded shafts in the Mathieu map

The equations of motion of an asymmetric Timoshenko shaft, that is having unequal principal moments of inertia, are derived within the framework of the Lagrangian formulation for continuous systems and fields. The Lagrangian density of the system is calculated in a moving frame, that is a rotating frame attached to the deformed shaft, and proves to depend on the four Lagrangian variables (fields) of the system and their first derivatives w.r.t. space and time.

Francesco Antonino Raffa

Papers

An analytical approach to the dynamics of rotating shafts

The dynamic stiffness method for linear rotor-bearing systems

Off-resonance regimes in nonlinear quantum Rabi models

Dynamic instability of axially loaded shafts in the Mathieu map

Equations of motion of an asymmetric timoshenko shaft