Advanced wave

One of the more recent approaches for designing xc functionals is based on inverting eq. (6.7). An accurate electron density may be calculated by advanced wave... 

The advanced wave is emitted backward in time as the retarded wave arrives at A and it arrives back at E at the instant that the retarded wave is being emitted. The two sites E and A may be astronomical distances apart and in the same way that simultaneity is not defined relativistically, the concept of instantaneous response also looses its meaning. The interaction between E and A is therefore non-local, irrespective of time differences. On absorption of the photon energy the wave function collapses everywhere. The photon represents the handshake between emitter and absorber, and therefore has no velocity. The constant c refers to the transmission of radiant energy between E and A, and the photon exists for the duration of the transmission. 

Closer scrutiny of the general wave equation provides an answer to the dilemma. As a second-order differential equation in the time variable, it has solutions in positive and negative time5. The negative time (or advanced) solutions are routinely rejected as physically impossible. This decision is based on prejudice rather than insight. Without evidence to accept only retarded (positive time) solutions as physically real, there is the possibility of response from a prospective receptor by means of advanced waves to establish one-to-one contact between emitter and absorber before transmission occurs. 

Suppose the vacuum to be filled with a uniform radiation field, or waves. Interaction of this field with an emitter, in an excited state, causes modulation of the wave field that spreads in the form of a spherical retarded wave. On reaching a suitable absorber, at a lower energy level, the modulation stimulates a matching sympathetic response, which is returned as an advanced wave, that reaches the emitter at the very time of initial modulation. Such a superposition of advanced and retarded waves amounts to the creation of a standing wave between emitter and absorber. Emitter and absorber are now in contact and the transaction is completed on the transfer of excess energy to the absorber. The standing wave that persists for the duration of the transaction triggers the transfer in the form of a photon. 

ON THE ADVANCED WAVE MODEL OF PARAMETRIC DOWN-CONVERSION... 

Abstract The spatiotemporal optical mode of the single-photon Fock state prepared by conditional measurements on a biphoton is investigated and found to be identical to that of a classical wave due to a nonlinear interaction of the pump wave and Klyshko s advanced wave. We discuss the applicability of this identity in various experimental settings. 

Furthermore, as demonstrated in our earlier theoretical work, the optical mode of the conditionally prepared photon is in fact completely identical to that of the difference-frequency field generated by a properly defined advanced wave [Aichele 2002], In other words, the advanced wave concept is not merely an informal visual tool, but a rigorous mathematical model which possesses analytic capability that approaches that of the canonical quantum theory. In the present paper, we review the theory associated with the advanced wave concept and discuss its applicability in various experimental situations. 

On the advanced wave model of parametric down-conversion... 

Here Ep(r, t) and EA(r, t) are the electric fields of the pump and advanced waves, respectively. The mode of the DFG field is obtained from Eq. (12) via a Fourier transform which is restricted to the crystal volume ... 

The advanced wave then enters the nonlinear crystal and interacts with the pump wave whenever and wherever it is present in the crystal. The nonlinear interaction of Klyshko s advanced wave with the pump pulse produces a pulse of DFG emission into the signal channel (Fig. 2 (a)). Substituting the correlation function (15) of the advanced wave into Eq. (14) as l we find that... 

The correlation function of the DFG pulse generated through the nonlinear interaction of the advanced wave and the pump pulse is identical to the density matrix of the single photon prepared by conditional measurements on a biphoton performed in the same optical arrangement.2... 

Unlike Klyshko, who said that the advanced wave is a -function pulse, we consider it to be a continuous, partially incoherent wave. The duration of the advanced wave is in fact determined by the uncertainty of the photon arrival time measurement. With modern detectors, it amounts to at least tens of picoseconds. If the down-conversion experiment is performed in an ultrashort pulsed setting, this uncertainty substantially exceeds the pump pulse width, so the advanced wave can be considered continuous. On the other hand, if the pump laser is continuous, the situation is more complicated and the timing uncertainty must be taken into account more rigorously in order to determine the correct correlation function of the DFG wave and the density matrix of the conditional single photon. 

Same considerations are valid for spatial coherence. As the advanced wave passes through a narrow aperture, it gains some degree of transverse coherence according to the van Cittert-Zernike theorem. Because the nonlinear interaction is restricted to the area where the pump field is present, the resulting signal (DFG) field is also partially coherent provided the pump beam diameter is smaller than the coherence width of the advanced wave in the plane of the crystal [Aichele 2002],... 

The basis sets described above are small and intended for qualitative or semiquantitative, rather than quantitative, work. They are used mostly for simple wave functions consisting of one or a few Slater determinants such as the Hartree-Fock wave function, as discussed in Sec. 3. For the more advanced wave functions discussed in Sec. 4, it has been proven important to introduce hierarchies of basis sets. New AOs are introduced in a systematic manner, generating not only more accurate Hartree-Fock orbitals but also a suitable orbital space for including more and more Slater determinants in the n-electron expansion. In terms of these basis sets, determinant expansions (Eq. (14)) that systematically approach the exact wave function can be constructed. The atomic natural orbital (ANO) basis sets of Almlof and Taylor [23] were among the first examples of such systematic sequences of basis sets. The ANO sets have later been modified and extended by Widmark et al. [24],... 

Due to the fact that on region I we have both incident and reflected wave (as in classical picture), in region II the classically forbidden behavior may include stationary waves back and forth inside the barrier, wile the region III being without wave source at infinity hosts only the transmitted (advanced) wave coming from tunneling region II. 

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