Collective quantization

The quantum-controlled NOT gate with the trapped ions can be implemented by exciting the collective quantized motion of the ions with lasers. The coupling of the motion of the ions is provided by the coulombic repulsion which is the dominant interaction for small ion-ion separations of the order of a wavelength. The final read-out of the quantum register (state measurement of the individual qubits) at the end of the computation can be accomplished using quantum jumps with unit efficiency [107, 108]. 

The limits of collective quantization in atoms are as yet quite unknown. In light atoms such as He or Be, the shell structure corresponding to principal quantum numbers is clearly marked even though the quantization corresponds to collective rather than independent-particle behavior. In heavy atoms such as Sr and Ba, the shell structure of the valence electrons seems to be blurred because excited states associated with one set of principal quantum numbers... 

Fracton A collective quantized vibration on a substrate with a fractal structure Free-dimensional space art The author I. Michaloudis is playing with the three-dimensional space which in the case of the indefinitive transparency of his aer( ) sculptures becomes a non-Euclidean space. Silica aerogel itself can be considered as a personification of what the French mathematician Henri Poincarre named a representative space, a space you cannot measure you just live in with all your senses for more information see loannis MICHALOU(di)S, Aer( jsculpture the enigmatic beauty of aerogel s nonentity in a pilot art and science project, Journal of Non-CrystaUine Sohds (2004) 350 61-66... 

A chemical reaction can be viewed as occurring via the formation of an excited state that can be any one of the degrees of freedom of the collection of N atoms. That is, translational, rotational, vibrational, and electronic excitation can lead to a chemical reaction. We often do not need to consider explicitly the quantized nature of rotational and vibrational energies in practical applications because of time scale considerations. For example, when a chemical reaction proceeds via a vibrationally excited state, in which the average lifetime typically is about 3 x 10" where T is in Kelvins... 

Ions in the lattice of a solid can also partake in a collective oscillation which, when quantized, is called a phonon. Again, as with plasmons, the presence of a boundary can modify the characteristics of such lattice vibrations. Thus, the infrared surface modes that we discussed previously are sometimes called surface phonons. Such surface phonons in ionic crystals have been clearly discussed in a landmark paper by Ruppin and Englman (1970), who distinguish between polariton and pure phonon modes. In the classical language of Chapter 4 a polariton mode is merely a normal mode where no restriction is made on the size of the sphere pure phonon modes come about when the sphere is sufficiently small that retardation effects can be neglected. In the language of elementary excitations a polariton is a kind of hybrid excitation that exhibits mixed photon and phonon behavior. 

A collection of studies on size quantization in CD CdSe films, together with some relevant data, is presented in Table 10.1. 

Exercise. The quantized harmonic oscillator in example (i) of VI.4 can be regarded as a collection of photons, each of which is either bound or free. Why is it nevertheless not possible to interpret the M-equation (VI.4.1) in the way described here ... 

Nuclei with certain even numbers of protons and of neutrons are the most stable. One model of the nucleus is the collective model, which pictures nucleons as occupying quantized energy levels and interacting with one another strongly. 

These are quantized vibrations of conduction electrons in a metal or semiconductor 4°). The quantized energy levels of the collective longitudinal vibrations of the electron gas are quasi-particles in the sense of the definition given above and are called plasmons. The frequency of this longitudinal vibration, the plasma frequency u>P, is given by 40>... 

Mossbauer s discovery [49] consisted in the fact that when the nuclei of the emitter and the absorber are included in a solid matrix, they vibrate in a crystal lattice [49,54,56], Therefore, owing to the essential quantum character of solid vibrations (see Section 1.4), the atoms located in a solid matrix are limited to a certain collection of quantized lattice vibration energies [54], Consequently, if the recoil energy is smaller than the lowest quantized lattice vibration energy, Ew, then / v = 0D, in which, k is the Boltzmann constant and 0D is the Debye temperature of the solid. In this case, this... 

The gap between mathematical theories and their applications, which is the theme of all the foregoing quotes, arises from one common source the discrepancy between the precision required by classical set theory (and its associated logic) and the inherent resolution limits of our perceptual capabilities as well as measuring instruments. Consider, for example, measurements of a physical quantity taken by a particular instrument. Due to the finite resolution of the instrument employed, appropriate quantization of the measurement is inevitable. Assume, for example, that the considered range of the quantity is [0,1] and that the measuring instrument allows us to measure with the accuracy of one decimal digit. Then, measurements are values taken from the collection of values 0, 0.1, 0.2,.. ., 0.9, 1, which stand for the intervals [0,0.05), [0.05,0.15),...,[0.85,0.95), [0.95,1]. This example of the usual quantization is illustrated in Fig. 4a. 

Physical chemists are well aware of the usefulness of models. An understanding of the fundamental properties of matter can hardly be gained from watching reality, requiring instead the posing of if-then questions that can be answered only by models. The nature of pressure or temperature of a gas as a collective property of its individual atomic or molecular constituents became obvious only through the billiard ball models of Clausius, Maxwell, and Boltzmann, despite our later insights that true atoms or molecules have quantized motion. 

As a consequence of the collective motion of the neutral system across the homogeneous magnetic field, a motional Stark term with a constant electric field arises. This Stark term inherently couples the center of mass and internal degrees of freedom and hence any change of the internal dynamics leaves its fingerprints on the dynamics of the center of mass. In particular the transition from regularity to chaos in the classical dynamics of the internal motion is accompanied in the center of mass motion by a transition from bounded oscillations to an unbounded diffusional motion. Since these observations are based on classical dynamics, it is a priori not clear whether the observed classical diffusion will survive quantization. From both the theoretical as well as experimental point of view a challenging question is therefore whether quantum interference effects will lead to a suppression of the diffusional motion, i.e. to quantum localization, or not. 

The collective model, in which nucleons are considered to occupy quantized energy levels and to interact with each other by the strong force and the electrostatic (coulomb) force... 

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