Superposition state dipole

Here 7 is the average radiative line half-width at half maximum (HWHM) of bound levels (which has been introduced phenomenologically), a>m = (Em — En)/ti, and < = (EM h Em). In obtaining this result, we have assumed that (a) the cw fields are turned on at t — —00, at which time the system is in its initial superposition state [Eq. (6.14)], (b) the medium has no permanent dipole moment, and (c) (El tl ZT2) = 0 due to the fact that IE)) and E2) are assumed to have the same parity. 

The bichromatic off-resonance LIP obeys the same general relation to the field as does the monochromatic LIP, namely, AW(y) = —dind E(y, f). All that one need do is calculate the dipole induced in the material superposition state by the bichro- matic field. Following our discussion of the control of refractive indices in Section, 6.2, the induced dipole is given by... 

The effect of quantum interference on spontaneous emission in atomic and molecular systems is the generation of superposition states that can be manipulated, to reduce the interaction with the environment, by adjusting the polarizations of the transition dipole moments, or the amplitudes and phases of the external driving fields. With a suitable choice of parameters, the superposition states can decay with controlled and significantly reduced rates. This modification can lead to subnatural linewidths in the fluorescence and absorption spectra [5,10]. Furthermore, as will be shown in this review, the superposition states can even be decoupled from the environment and the population can be trapped in these states without decaying to the lower levels. These states, known as dark or trapped states, were predicted in many configurations of multilevel systems [11], as well as in multiatom systems [12],... 

The CPT effect and its dependence on quantum interference can be easily explained by examining the population dynamics in terms of the superposition states. v) and a). Assume that a three-level A-type atom is composed of a single upper state 3) and two ground states 1) and 2). The upper state is connected to the lower states by transition dipole moments p31 and p32. After introducing superposition operators 5+ = (S ) = 3)(.v and 5+ = (Sa) = 3)(a, where. v) and a) are the superposition states of the same form as Eqs. (107) and (108), the Hamiltonian (65) can be written as... 

As we have shown in Sec. III. A, the second-order correlation function of the fluorescence field depends on correlation functions of the atomic dipole moments (S+(f)S+(f + x)Sy(t)Sj (t)), which correspond to different processes including photon emissions from a superposition of the excited levels. Therefore, we write the correlation functions G (R, t) and G (R, t R, t + x) in terms of the symmetric and antisymmetric superposition states as... 

It is easily to find from Eq (153) that the dipole matrix elements between the superposition states and the ground state 2) are... 

It is therefore possible to construct a superposition of states of the dipole, such that the effective dipole moment of the molecule bobs up and down in time. The amount that the dipole bobs, relative to the constant component, can be controlled by the relative population in the two states. Moreover, the degree of polarization of the molecule plays a significant role. For a fully polarized molecule, when 81/2 = 0, only the dc portion of the dipole persists, although even it can vanish if there is equal population in the two states, dipole up and dipole down. ... 

As before, the molecule can also be in a superposition state. No matter how complicated this superposition is, the expectation value of the dipole must instantaneously point in some direction, since the only available angular dependence resides in the C q functions, which yield only dipoles. In other words, no superposition can generate the field pattern of a quadrupole moment, for example. 

To see that the dipole moment of the superposition state oscillates with time, consider the same states at time t = (1 /2)h/ Eb — Ea). For the particle in a box, the energy of the second eigenstate is four times that of the first, Eb = AEa (Eq. 2.24), so ll2)hl Eb — Ea) = /6)h/Ea- The individual wavefunctions at this time have both real and imaginary parts. For the lower-energy state, we find by using the relationship exp(—/0) = cos (0) — i sin (0) that... 

This differs from the superposition state at zero time in that it depends on the difference between t/ aCx) and instead of the sum (Fig. 4.3B). Inspection of the electron density function in Fig. 4.3 > shows that the electric dipole of the superposition state has reversed direction relative to the orientation at... 

We can relate the amplitude of the oscillating dipole of the superposition state to the transition dipole (/ /, ) as follows. For a superposition state CfllPa -b Cb Pb with P = il)i o —iEkt/K), the expectation value of the dipole is... 

State I ) m the electronic ground state. In principle, other possibilities may also be conceived for the preparation step, as discussed in section A3.13.1, section A3.13.2 and section A3.13.3. In order to detemiine superposition coefficients within a realistic experimental set-up using irradiation, the following questions need to be answered (1) Wliat are the eigenstates (2) What are the electric dipole transition matrix elements (3) What is the orientation of the molecule with respect to the laboratory fixed (Imearly or circularly) polarized electric field vector of the radiation The first question requires knowledge of the potential energy surface, or... 

The electron-phonon operator is a tensor product between the electronic dipole and the nuclear dipole operators. A mixing between the AA and BB via the singlet-spin diradical AB state is possible now. A linear superposition of identical vibration states in AA and BB is performed by the excited state diradical AB. If the system started at cis state, after coupling may decohere by emission of a vibration photon in the trans state furthermore, relaxation to the trans... 

Stockmayer potential is considered as a superposition of a Lennard-Jones (6-12) potential and the interaction of two point dipoles. Many of the properties of gases and liquids have been calculated in terms of these two potential functions. It should be borne in mind, however, that Lennard-Jones and Stockmayer potentials are idealizations of the true energy of interaction and that they are reasonably accurate for a number of simple molecules. The interaction of long molecules, molecules in excited states, free radicals, and ions cannot be described by these two potential functions (Ref 8a, pp 23 35)... 

Note that the above equations for the radiation of a dipole refer to a single dipolar source of radiation. For our spectroscopic interests, we more commonly are dealing with an assembly of many identical such systems. Usually, the total radiation of such an assembly of sources may be assumed to be the incoherent superposition of the individual intensities (see, however, the discussions of the intercollisional effect below, Chapters 3 and 5). We may then simply multiply the contributions of one source by the number iV of sources in the (initial) state i. 

The generalization to the control of the dynamics of a molecule with n electronic states is straightforward. For the purpose of deducing the control conditions we will examine the extreme case in which every possible pair of these electronic states is connected via the radiation field and a nonzero transition dipole moment. If the molecule is coupled to a radiation field that is a superposition of individual fields, each of which is resonant with a dipole allowed transition between two surfaces, the density operator of the system can be represented in the form... 

Using this approach the +) and —) states are not coupled by the field of the ion, but are only split in energy. At high collision velocities the initial state 0) is simply projected onto the 0 + 1) state, a coherent superposition of +) and -) states, by the dipole matrix element. However, at lower velocities the change in energy of the +) and -) states during the collision allows the +) and -) states themselves to be populated rather than only a coherent superposition. The latter feature allows nondipole transitions at lower collision velocities, as observed experimentally. 

Turning to the barriers we note that they are defined as the difference in enthalpy between the transition state and the separated reactants. This is important because in gas-phase ion-molecule reactions the transition state is typically preceded by an ion-dipole complex131 133 formed between the reactants, and the term barrier is sometimes used for the enthalpy difference between the transition state and the ion-dipole complex. However, these ion-dipole complexes have little relevance to the main topic discussed in this chapter and hence the chosen definition of AH is more appropriate. For reasons explained elsewhere,118 the barriers reported in Table 16 have not been corrected for the basis set superposition error (BSSE),134 although such corrected values are available.118... 

Equation (2.35)]. Equation (15.6b) is formally equivalent to (2.66) with the exception that in the present case the outgoing channel also includes, in addition to the vibrational state, the particular electronic state. It is important to realize that because of the nonadiabatic coupling both excited electronic states and both electronic product channels are populated, even if one transition dipole moment is exactly zero for all nuclear geometries. Furthermore, the superposition of two complex-valued amplitudes in the case that both transition moments are non-zero can lead to interesting interference patterns. 

Pendular state Superpositions of field-free rotational eigenstates in which the molecular axis librates about the field direction. Pendular states are eigenstates of the rotational Hamiltonian plus the dipole potential. 

Finally, we note that a number of experiments have shown that it is possible to. accelerate or decelerate molecules using time-varying electric fields [244], In this case the molecule is passed through an array of synchronously pulsed electric field stages that interact with the molecular dipole. Since the dipole is a function of the "J state of the system, it may be possible to prepare a superposition of internal states and then selectively accelerate one of the two internal states to produce the desired J superposition.. 1... 

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