Continuum degenerate

References 29-33 introduce the notion of coherence spectroscopy in the context of two-pathway excitation coherent control. Within the energy domain, two-pathway approach to coherent control [25, 34—36], a material system is simultaneously subjected to two laser fields of equal energy and controllable relative phase, to produce a degenerate continuum state in which the relative phase of the laser fields is imprinted. The probability of the continuum state to evolve into a given product, labeled S, is readily shown (vide infra) to vary sinusoidally with the relative phase of the two laser fields < ),... 

Comprehensive experimental investigations of 2PA processes in fluorene derivatives were performed by Hales et al. [53,56-59] with open aperture Z-scan [26], two-photon induced fluorescence [60] and femtosecond white-light continuum pump-probe methods [61]. For degenerate two-photon excitation, the experimental 2PA spectra of symmetrical and asymmetrical fiuorenes are presented in Figs. 15 and 16. These spectra were obtained with the combination of open aperture Z-scan and two-photon fluorescence methods [57]. For centrosymmetric molecules, two-photon transitions from... 

The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a,. Thus, varying the coefficients a, allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. 

The time evolution is determined by the full effective Hamiltonian H and not by the rate matrix T alone. One cannot therefore discuss the time evolution without reference to the matrix H. Say, however, H and T commute, [H, T] = 0. A simple condition that ensures this result is that the bound states are strictly degenerate. If H and T commute, the eigenvectors of T evolve in time independently of one another. In the basis of states defined by the N eigenvectors of T there will be K states that will decay by direct coupling to the continuum and N - K states that are trapped forever. An arbitrary initial state is a linear combination of the N eigenvectors of T and hence can have a trapped component. 

The most important implication of not being a good quantum number is that blue and red states are coupled by their slight overlap at the core. In the region below the classical ionization limit blue and red states of adjacent n do not cross as they do in H, but exhibit avoided crossings as a result of their being coupled. Above the classical ionization limit blue states, which would be perfectly stable in H, are coupled to degenerate red states, which are unbound, and ionization occurs rapidly compared to radiative decay. It is really an autoionization process in which the blue state is coupled to the red continuum state at the ionic core. 

Since Vd(r) is only nonzero near r = 0 the matrix element of Eq. (6.51) reflects the amplitude of the wavefunction of the continuum wave at r 0. Specifically, the squared matrix element is proportional to C, the density of states defined earlier and plotted in Fig. 6.18. From the plots of Fig. 6.18 it is apparent that the ionization rate into a continuum substantially above threshold is energy independent. However, as shown in Fig. 6.18, there is often a peak in the density of continuum states just at the threshold for ionization, substantially increasing the ionization rate for a degenerate blue state of larger This phenomenon has been observed experimentally by Littman et al.32 who observed a local increase in the ionization rate of the Na (12,6,3,2) Stark state where it crosses the 14,0,11,2 state, at a field of 15.6 kV/cm, as shown by Fig. 6.19. In this field the energy of the... 

The autoionizing states of channel 2 are represented by Fig. 21.3, however they do not necessarily appear as in Fig. 21.3 in a photoionization spectrum. Let us first consider photoexciting the autoionizing states of channel 2 and the degenerate continuum of channel 1 from a compact initial state, g, such as the ground state. Since the initial state is spatially localized near the ionic core, only the part of the Rydberg wavefunction near the core plays an active role in the excitation. We can write the dipole matrix element for the excitation in either of two ways,1-3... 

In Fig. 21.10 we show the 6pnd states as being degenerate with only a single continuum, above the 6s1/2 limit.15 This simplifying assumption allows us to treat the problem with only three channels. It is important to recall that in recording an ICE spectrum we do not directly excite the continuum, it acts only as a sink for electrons. As a result, the fact that the continuum is not well characterized is not important. 

Fig. 21.10 The 6pnd 7 = 3 levels of Ba converging to the 6pI/2 and t irl limits. Note that the 6p3/2 lOdj state is degenerate with the 6pi/220d5/2 state. The 6p3/2lldj and higher states are above the 6p1/2 limit. All the 6pnd states lie in the continuum above the Ba+ 6s1/2 state...

The system of integral equations [Eq. (66)] is eventually discretized and solved with numerical linear algebra procedures. At each energy, the system (66) must be solved for each of the open channels. A complete set of linearly independent degenerate real (i.e., stationary) continuum solutions if"E is thus obtained. The stationary scattering states xjr E are not orthogonal it can be shown that their superposition is given by... 

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

In Section 2.5 we have constructed the degenerate continuum wavefunc-tions 4/ f(R, r Ef, n), which describe the dissociation of the ABC complex into A+BC(n). They solve the time-independent Schrodinger equation for fixed energy Ef subject to the boundary conditions (2.59). Furthermore, the 4/f(R,r Ef,n) are orthogonal and complete and thus they form a basis in the corresponding Hilbert space, i.e., any function can be represented as a linear combination of them. 

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