Coherent state transition amplitude

To discuss the semiclassical spin-coherent state propagator, we consider a general transition amplitude ( / e which can be expressed as an integral... 

We will determine a coherent-state/holomorphic path integral representation for the parallel-transport operator, deriving an appropriate transition amplitude [a path integral counterpart of the l.h.s. in Eq. (14)]. 

From a frequency domain point of view, a femtosecond pump-probe experiment, shown schematically in Fig. 1, is a sum of coherent two-photon transition amplitudes constrained by the pump and probe laser bandwidths. The measured signal is proportional to the population in the final state Tf) at the end of the two-pulse sequence. As these two-photon transitions are coherent, we must therefore add the transition amplitudes and then square in order to obtain the probability. As discussed below, the signal contains interferences between all degenerate two-photon transitions. When the time delay between the two laser fields is varied, the... 

Figure 1. The creation, evolution, and detection of wave packets. The pump laser pulse pump (black) creates a coherent superposition of molecular eigenstates at t — 0 from the ground state I k,). The set of excited-state eigenstates N) in the superposition (wave packet) have different energy-phase factors, leading to nonstationary behavior (wave packet evolution). At time t = At the wave packet is projected by a probe pulse i probe (gray) onto a set of final states I kf) that act as a template for the dynamics. The time-dependent probability of being in a given final state f) is modulated by the interferences between all degenerate coherent two-photon transition amplitudes leading to that final state.

If the photon pulse is sufficiently short and transitions from one initial J", Mj state to two or more excited eigenstates belonging to different J (same-Mj) quantum numbers lie within the photon spectral width, then a nonstation-ary state is prepared at t = 0. If the coherently excited J = J", J" 1, Mj eigenstates are not degenerate, the transition amplitudes for the transitions into common eigenstates will interfere in a time-dependent manner. 

The transition amplitude for the dipole transition from the dark state ui)coherent to the upper state 3> is... 

The so-called coherent state representation is useful also in semiclassical calculations. One choice of the coherent states is due to Herman and Kluk t3rpe, provide that the relevant trajectories do not undergo chaotic behavior. The transition amplitude is expressed by... 

We have already discussed quantum-beat spectroscopy (QBS) in connection with beam-foil excitation (Fig.6.6). There the case of abrupt excitation upon passage through a foil was discussed. Here we will consider the much more well-defined case of a pulsed optical excitation. If two close-lying levels are populated simultaneously by a short laser pulse, the time-resolved fluorescence intensity will decay exponentially with a superimposed modulation, as illustrated in Fig. 6.6. The modulation, or the quantum beat phenomenon, is due to interference between the transition amplitudes from these coherently excited states. Consider the simultaneous excitation, by a laser pulse, of two eigenstates, 1 and 2, from a common initial state i. In order to achieve coherent excitation of both states by a pulse of duration At, the Fourier-limited spectral bandwidth Au 1/At must be larger than the frequency separation ( - 2)/ = the pulsed excitation occurs at... 

Emission processes always lead to a photon-related recoil velocity for an atom. Thus, in the quest for stiU lower temperatures it is necessaury to ascertain that an atom, which for some reason is brought to a standstill, can be exempt from further interaction with the fight. This is possible if the atom is placed in a so-called dark state [9.440]. If the atom is in a coherent superposition of two ground-state sublevels, from which the transition amplitudes exhibit a total destructive interference, a dark state is achieved (See also Sect. 9.5.3). It can be shown that for counter-propagating beams with circular polariza-... 

Next we will examine possible origins of quantum interference in 2PP via the Cs a-resonance on the Ag(lll) surface. Quantum interference occurs when a transition from the same initial and final states can occur coherently via multiple excitation pathways. When taking the coherent sum over all pathways, phase differences cause constructive and destructive interferences between different amplitudes. Accounting only for the resonant interactions, the photoemitted intensity of the 2PP process depends on one-photon transition amplitudes promoting an electron first from an initial state to two possible, intermediate... 

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

The first event may happen anywhere on the TV screen you can prepare the system as many times as you want and check that the first event appears localized (almost) at random this randomness is only apparent if you use the theory presented here. What has happened was a change in amplitudes for a transition from state +) to —) by capturing energy from the I-frame system the relative coherent intensity response being ... 

UPS from Adsorbate Core Levels.—As outlined above, an out-going photoelectron in its final state is a super-position of two coherent contributions a direct wave whose amplitude and symmetry are determined by the intra-atomic transition at the emitting site and an indirect wave generated by repeated scattering of the direct wave by the local atomic environment. It was suggested by Liebsch that this final-state scattering should lead to angular variations in the photoemission spectrum and would be examined best in core-level emission, which involves the simplest possible initial... 

A more direct evidence of the surface localized excitation mechanism has been obtained by a polarization dependence study. For K/Pt(lll) at 0.36 ML, it has been demonstrated that the coherent excitation of the K—Pt stretching mode occurs with p-polarized excitation and not with s-polarized exdtation. Since the s-polarization absorptance is about one fourth of that with p-polarization under the experimental conditions (2.19 eV photon energy, 70° angle of inddence), the coherent amplitude should be detectable with s-polarization if the substrate-mediated process operates. Therefore, the negligible oscillatory component with s-polarization is inconsistent with the substrate-mediated excitation model and it is indicated that some electronic transitions involving K-induced surface states are responsible for the coherent excitations. 

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