Zero-point fluctuations

The origin of this force can be traced back to the zero-point fluctuations of the electromagnetic field which are modified by the addition of the two plates relative to the free case. This induces the following change in the energy of the vacuum ... 

Finally, in the quantum approximation the radiation is no longer treated classically (i.e., using Maxwell s equation), and so both radiation and matter are described by quantum methods. For most of the features in the spectra of solids, this approach is not necessary and it will not be invoked. However, this approximation also leads to important aspects, such as zero-point fluctuations, which are relevant in the theory of lasers and Optical Parametic Oscillators (Chapter 3). 

These methods have been applied to calculate the polarizabilities of atoms,31 and the long-ranged forces between atoms,33 with a typical calculated accuracy of 10 % or less. Thus, we have been able to estimate successfully the significant features of zero-point fluctuations of atomic dipole moments, without actually solving the quantum equations of motion to obtain all the excited state energies and wave functions. 

The cohesion in crystals of electrically neutral atoms, typified in Figure 5.10 by Ar, is ascribed to van der Waals interaction, generated by zero-point fluctuation of the electron density and its polarization effect on neighbouring atoms. The energy of interaction is given by the London formula, as a function of zero-point vibration frequency, between atoms at a distance d apart, as ... 

A feature of London s paper is its emphasis on the zero-point motion of electrons it is the intermolecular correlation of this zero-point motion that is responsible for dispersion forces. London s Section 9 extends the idea of zero-point fluctuations to the interaction of dipolar molecules. If their moment of inertia is small, as it is for hydrogen halide molecules, then even near the absolute zero of temperature when the molecules are in their non-rotating ground states, there are large fluctuations in the orientation of the molecules and these become correlated in the interacting pair. 

The connection between the non-resonant nonlinear optical (NLO) response to optical pumping well below the absorption edge and the resonant NLO response following absorption and photo-excitation has been discussed in terms of contributions from virtual soliton pairs enabled by nonlinear zero-point fluctuations in the ground state [82,221,222]. Because of the nonlinear zero-point fluctuations, there are finite matrix elements connecting the ground state with the relaxed state following the creation of a soliton-antisoliton (S-AS ) pair. This... 

The importance of virtual solitons in the ground state to fran -polyacetylene nonlinear zero-point fluctuations, was subsequently developed in detail... 

Here Vhas the same meaning as in (21). Thus, the zero-point fluctuations of the electric field strength of plane waves have the same magnitude at any space point. By construction, (28) describes the zero-point fluctuations in empty space. 

The trace of (143) coincides, apart from an unimportant factor, with the zero-point fluctuations of energy (29). 

Unruh effect The phenomenon, predicted in 1976 by the Canadian physicist William Unruh, that an accelerating body would seem to be surrounded by particles at a non-zero temperature, which is proportional to the acceleration. The vacuum state of a non-accelerating observer is different to that of an accelerating observer because of distortion of the zero-point fluctuations. The effect itself is very smffil and has not been verified experimentally. There is Unruh radiation associated with this effect. The effect itself is very small and has not been verified experimentally. 

Fig. 9.97 Uncertainty areas for different squeezing conditions (a) ( i) = 0, A 2 is squeezed but A(p is larger than in the nonsqueezed case (b) general case of squeezing of A at the expense of increasing A0 (c) uncertainty area of zero-point fluctuations with ( i) = ( 2) = 0 (d) minimized A(f) at the expense of larger AE...

If the incident laser beam in Fig. 9.96a is blocked, the mean intensity (/) becomes zero. However, the measured noise power density Pn(/) does not go to zero but approaches a lower limit po that is attributed to the zero-point fluctuations of the vacuum field, which is also present in a dark room. The interferometer in Fig. 9.96a has two inputs the coherent light field and a second field, which, for a dark input part, is the vacuum field. Because the fluctuations of these two inputs are uncorrelated, their noise powers add. Increasing the input intensity Iq will increase the signal-to-noise-ratio... 

A particular general shortcoming of the alloy analogy approximation is that it cannot describe quantum fluctuations such as the zero point fluctuations of spin waves. Evidently, these can be important at and near T = 0. Moreover, an ensemble average of static fluctuations depicted by the alloy configurations will, within the CPA, inevitably lead to quasi-particles with finite lifetime even at T = 0. Namely, the ground state is genetically not that of a Fermi liquid as it mostly should be. In what follows we shall summarize briefly the current state of conceptual framework that needs to be invoked to deal with these issues. 

This effect may be explained in terms of the quantum theory of electrodynamics. According to this theory each mode of the quantized radiation field possesses a zero-point energy of hu)/2. This implies that, even in the absence of external radiation, the mean square value of the time-depen-dent electric field is finite and that a hydrogen atom will experience a perturbation produced by the fluctuations in this field. These zero-point fluctuations cause the electron to wobble randomly in its orbit and so smear the charge over a greater volume of space. Since the electron is bound to the nucleus by a non-uniform electric field, the reduction in electron density causes a shift in the atomic energy levels. This Lamb shift, as it is now called, is greatest for those states in which iK0) is finite, i.e. the n states. 

The relaxation processes. The second perturbation term in equation (15.16) represents the effect of relaxation processes. At present we are concerned only with the excited-state density matrix and assume that the relaxation is due only to spontaneous emission. Since this is again an essentially random process, being triggered by the zero-point fluctuations of the vacuum radiation fields, the effect of can be represented as a rate process and we have... 

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