Continuum superposition state

In this paper we generalize our previous results and show that direct as well as resonant processes can be controlled via the coherent generation of a degenerate continuum superposition state. Moreover we show that decay via a resonance (predissociation) enhances the probability of full control. The result is a practical way of altering product ratios in chemical (e.g. dissociation and ionization) decay processes using (weak) coherent radiation. 

Section II of this paper contains the theory of the preparation and decay of a continuum superposition state and a theoretical discussion on the limits of control attained. In section III we present a number of computational results in which we demonstrate how to achieve control over the yield of I vs. I in the photodissociation of a diatomic (FI) and a polyatomic (CHo I) molecule. We also show how a single vibronic state of a photo-fragment can be enhanced or completely turned off. 

II. Preparation and Decay of A Continuum Superposition State A. General Formulation... 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

In a time-dependent picture, resonances can be viewed as localized wavepackets composed of a superposition of continuum wavefimctions, which qualitatively resemble bound states for a period of time. The unimolecular reactant in a resonance state moves within the potential energy well for a considerable period of time, leaving it only when a fairly long time interval r has elapsed r may be called the lifetime of the almost stationary resonance state. 

Figure 1. Schematic illustration of two-pathway control in the (a) frequency and (b) time domains. In case (a) the ground state is excited to a coupled continuum by either one photon of frequency CO3 or three photons of frequency C >i. Control is achieved by introducing a phase lag between the two fields. In case (b) a two-pulse sequence has sufficient bandwidth to excite a superposition of two intermediate states. Control is achieved by introducing a delay, At, between the pulses, resulting in a phase difference of to At.

Considering again the case of a structureless continuum, we have that 8j3 arises from excitation of a superposition of continuum states, hence from coupling within PHmP [69]. The simplest model of this class of problems, depicted schematically in Fig. 5b, is that of dissociation of a diatomic molecule subject to two coupled electronic dissociative potential energy curves. Here the channel phase can be expressed as... 

One thus identifies the adiabatic states of 9.33, with the 0 ) bound states of a Q space within the PT projection formalism given in Section 9.2 these state become resonances, since they are superpositions of the I q) and IA2) states, which interact with continuum states c,E 1) in the P space. Following Section 9.2, it follows that the E - QhQ matrix in such a case of two overlapping resonances case is given by... 

Figure 4. Schematic of the potential energy curves of the relevant electronic states The pump pulse prepares a coherent superposition of vibrational states in the electronic A 1 EJ state at the inner turning point. Around v = 13 this state is spin-orbit coupled with the dark b 3n state, causing perturbations. A two-photon probe process transfers the wavepacket motion into the ionization continuum via the (2) llg state [7].

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. 

Let us start with zero approximation states of H0 consisting of the discrete states (Xx, X2), 2(Xls X2),..., n( -i, X2) and continuum states time-independent) eigenstates of the physical system are obtained by diagonalizing the total Hamiltonian in this representation, and can be displayed as a superposition of these zero-order states. For the sake of simplicity we consider just one zero-order... 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

These are produced by autoionization transitions from highly excited atoms with an inner vacancy. In many cases it is the main process of spontaneous de-excitation of atoms with a vacancy. Let us recall that the wave function of the autoionizing state (33.1) is the superposition of wave functions of discrete and continuous spectra. Mixing of discrete state with continuum is conditioned by the matrix element of the Hamiltonian (actually, of electrostatic interaction between electrons) with respect to these functions. One electron fills in the vacancy, whereas the energy (in the form of a virtual photon) of its transition is transferred by the above mentioned interaction to the other electron, which leaves the atom as a free Auger electron. Its energy a equals the difference in the energies of the ion in initial and final states ... 

Conversely, a coherent superposition of continuum states with a population closely reproducing an isolated peak in the density of states, which corresponds to a resonance, can be built in such a way to give rise to a localized state. From this localized state, there will be an outward probability density flux, i.e., it will have a finite lifetime. In the limit of a resonance position far from any ionization threshold and a narrow energy width, the decay rate will be exponential with the rate constant T/ft. The decay is to all the available open channels, in proportion to their partial widths. 

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... 

The essential difference between the multistate and the continuum model rests upon the temperature dependence of the density of states (Fig. 13). An experimental distinction between the models may also be possible (21) from a study of genuine vitreous (quick-frozen) solutions, i.e., systems that, at low temperature, retain the appropriate fluid solution species existing at the freezing point. For these systems, the continuum model predicts a single species at low temperature with a low A value, while the multistate picture requires the superposition of spectra from two (or more) species one (or more) in high abundance with a low A value, and at least one in lower abundance with a high value of A (Fig. 13). 

Resonantly enhanced two-photon dissociation of Na2 from a bound state of the. ground electronic state occurs [202] by initial excitation to an excited intermediate bound state Em,Jm, Mm). The latter is a superposition of states of the A1 1+ and b3Il electronic curves, a consequence of spin-orbit coupling. The continuum states reached in the two-photon excitation can have either a singlet or a triplet character, but, despite the multitude of electronic states involved in the computation reported J below, the predominant contributions to the products Na(3s) + Na(3p) and Na(3s) + Na(4s) are found to come from the 1 flg and 3 + electronic states, respectively. The resonant character of the two-photon excitation allows tire selection of a Single initial state from a thermal ensemble here results for vt = Ji — 0, where vt,./, denote the vibrational and rotational quantum numbers of the initial state, are stJjseussed. 

Fig. 8.2 Schematic showing controlled dissociation of molecule B-A-B to yield B-A+B, or B + A-B products, where B and B are two enantiomers. A molecule is excited from ah 3 initial state i i) to a superposition of antisymmetric (] 2)) and symmetric (]/ 3)) vibratiorialf states belonging to an excited electronic state, by excitation pulse x(a>). After an appropriate delay time, the molecule is dissociated by second pulse ed(ca), to the E, n, D ) or E, n, continuum state. fv I ...

Thus the excitation pulse can create a superposition of i), 2) consisting of two states of different reflection symmetry. The resultant superposition possesses no symmetry properties with respect to reflection [78]. We now show that the broken symmetry created by this excitation of nondegenerate bound states translates into a nonsymmetry in the probability of populating the degenerate , n, D ), , n, L ) continuum states upon subsequent excitation. To do so we examine the properties of the bound-free transition matrix elements ( , n, q de,g Ek) that enter into the probability of dissociation. Note first that although the continuum states , n, q ) are nonsymmetric with respect to reflection, we can define symmetric and antisymmetric continuum eigenfunctions , n, s ) and , n, a ) via the relations... 

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