Electron phase coherence

Let us finally discuss to what extent the MFT method is able to (i) obey the principle of microreversibility, (ii) account for the electronic phase coherence, and (iii) correctly describe the vibrational motion on coupled potential-energy surfaces. It is a well-known flaw of the MFT method to violate quantum microreversibility. This basic problem is most easily rationalized in the case of a scattering reaction occurring in a two-state curve-crossing system, where the initial and final state of the scattered particle may be characterized by the momenta p, and pf, respectively. We wish to calculate the probability Pi 2 that... 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

D. J. Tannor I would like to point out that the Scherer-Fleming wavepacket interferometry experiment is very different from the Tannor-Rice pump-dump scheme, in that it exploits optical phase coherence of the laser light (optical phase coherence translates into electronic phase coherence between the wavepackets on different potential surfaces). However, there was a paragraph in the first paper of Tannor and Rice [7. Chem. Phys. 83, 5013 (1985), paragraph above Eq. (11)] that did in fact discuss the role of optical phase and suggested the possibility of experiments of the type performed by Scherer and Fleming. 

Measurements of electronic phase coherence and its decay are discussed in the next section. Pump-probe experiments with two incident laser pulses and... 

When the frontier orbitals of the bridge are not much higher in energy than those of D (for ET) or A (for HT), the dominant mechanism becomes incoherent charge hopping. Figure 4.16b shows ET by this mechanism. The electron hops in a thermally activated step from D onto the bridge frontier orbital(s) before it comes to rest on A. The intermediate state D B A is populated and the electron phase coherence is lost. The... 

The remaining sections are devoted to a discussion of the various MQC methods. Among other issues, we consider the abihty of a method to (i) account for the branching of trajectories, (ii) accoimt for the electronic phase coherence, (iii) correctly describe the vibrational motion on coupled... 

Static defects scatter elastically the charge carriers. Electrons do not loose memory of the phase contained in their wave function and thus propagate through the sample in a coherent way. By contrast, electron-phonon or electron-electron collisions are inelastic and generally destroy the phase coherence. The resulting inelastic mean free path, Li , which is the distance that an electron travels between two inelastic collisions, is generally equal to the phase coherence length, the distance that an electron travels before its initial phase is destroyed ... 

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

The terms elastic and inelastic scattering of electrons describe that which results in no loss of energy and some measureable loss of energy respectively. If the incident electron beam is coherent (i.e. the electrons are in phase) and of a fixed wavelength, then elastically scattered electrons remain coherent and inelastic electrons are usually incoherent. 

In ESR, it is also customary to classify relaxation processes by their effects on electron and nuclear spins. A process that involves an electron spin flip necessarily involves energy transfer to or from the lattice and is therefore a contribution to Tx we call such a process nonsecular. A process that involves no spin flips, but which results in loss of phase coherence, is termed secular. Processes that involve nuclear spin flips but not electron spin flips are, from the point of view of the electron spins, nonsecular, but because the energy transferred is so small (compared with electron spin flips) these processes are termed pseudosecular. 

The Landauer formula assumes elastic processes. If the electrons move coherently (that is without any loss of energy or of phase) they will tunnel if the energy gap through which they must tunnel becomes relatively small, they can tunnel a long way. Generally, the conduction in the tunneling regime is written as... 

Landauer proposed in 1957 the first mesoscopic theoretical approach to charge transport [176]. Transport is treated as a scattering problem, ignoring initially all inelastic interactions. Phase coherence is assumed to be preserved within the entire conductor. Transport properties, such as the electrical conductance, are intimately related to the transmission probability for an electron to cross the system. Landauer considered the current as a consequence of the injection of electrons at one end of a sample, and the probability of the electrons reaching the other end. The total conductance is determined by the sum of all current-carrying eigenmodes and their transmission probability, which leads to the Landauer formula of a ID system ... 

Bauer R, Neuhauser D (2002) Phase coherent electronics a molecular switch based on quantum interference. J Am Chem Soc 124 4200... 

In order to obtain estimates of quantum transport at the molecular scale [105], electronic structure calculations must be plugged into a formalism which would eventually lead to observables such as the linear conductance (equilibrium transport) or the current-voltage characteristics (nonequilibrium transport). The directly measurable transport quantities in mesoscopic (and a fortiori molecular) systems, such as the linear conductance, are characterized by a predominance of quantum effects—e.g., phase coherence and confinement in the measured sample. This was first realized by Landauer [81] for a so-called two-terminal configuration, where the sample is sandwiched between two metalhc electrodes energetically biased to have a measurable current. Landauer s great intuition was to relate the conductance to an elastic scattering problem and thus to quantum transmission probabilities. 

The flow of energy from the ground state can also be calculated when it is assumed that the rate of electronic dephasing is small (i.e., L%Pg = 0) and/or the rate of pure vibrational dephasing is small (i.e., L%Hg = 0). These conditions apply when the rate of relaxation to equilibrium is small relative to the rate of loss of phase coherence. Under these conditions... 

Assume that a noninteracting nanosystem is coupled weakly to a thermal bath (in addition to the leads). The effect of the thermal bath is to break phase coherence of the electron inside the system during some time Tph, called decoherence or phase-breaking time. rph is an important time-scale in the theory, it should be compared with the so-called tunneling time - the characteristic time for the electron to go from the nanosystem to the lead, which can be estimated as an inverse level-width function / 1. So that the criteria of sequential tunneling is... 

The Cooper pairs are bosons, and below a critical Tc (which is affected by both applied pressure and by applied magnetic field) can condense to the same momentum state and wavefunction for all Cooper pairs in the solid these pairs have long-distance phase coherence and are present in all known superconductors. However, the condensation of these Cooper pair bosons is attributed to electron-phonon coupling only for monoatomic and diatomic metals (BCS theory), where the critical temperature Tc depends on isotopic mass. 

The X rays, which are produced with lack of phase coherence with intensity I0, if they impinge on a stationary electron at the origin, scatter (Thomson scattering) with intensity I at a distance R from the electron, at an angle 8 from the direction of the incoming beam as follows ... 

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