Superposition coherent

This section begins with a brief description of the basic light-molecule interaction. As already indicated, coherent light pulses excite coherent superpositions of molecular eigenstates, known as wavepackets , and we will give a description of their motion, their coherence properties, and their interplay with the light. Then we will turn to linear and nonlinear spectroscopy, and, finally, to a brief account of coherent control of molecular motion. 

Much of the previous section dealt with two-level systems. Real molecules, however, are not two-level systems for many purposes there are only two electronic states that participate, but each of these electronic states has many states corresponding to different quantum levels for vibration and rotation. A coherent femtosecond pulse has a bandwidth which may span many vibrational levels when the pulse impinges on the molecule it excites a coherent superposition of all tliese vibrational states—a vibrational wavepacket. In this section we deal with excitation by one or two femtosecond optical pulses, as well as continuous wave excitation in section A 1.6.4 we will use the concepts developed here to understand nonlinear molecular electronic spectroscopy. 

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

A microwave pulse from a tunable oscillator is injected into the cavity by an anteima, and creates a coherent superposition of rotational states. In the absence of collisions, this superposition emits a free-mduction decay signal, which is detected with an anteima-coupled microwave mixer similar to those used in molecular astrophysics. The data are collected in the time domain and Fourier transfomied to yield the spectrum whose bandwidth is detemimed by the quality factor of the cavity. Hence, such instruments are called Fourier transfomi microwave (FTMW) spectrometers (or Flygare-Balle spectrometers, after the inventors). FTMW instruments are extraordinarily sensitive, and can be used to examine a wide range of stable molecules as well as highly transient or reactive species such as hydrogen-bonded or refractory clusters [29, 30]. 

We use a7r/2 — vr — vr/2 pulse sequence to coherently divide, deflect and finally recombine an atomic wavepacket. The first vr/2 pulse excites an atom initially in the l,p) state into a coherent superposition of states l,p) and 2,p + hkeff). If state 2) is stable against spontaneous decay, the two parts of the wavepacket will drift apart by a distance hkT/m in time T. Each partial wavepacket is redirected by a vr pulse which induces the transitions... 

If the target is a pure electronic state (or coherent superposition state) in the conduction band, c)) the target operator can be written as... 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

It is usually assumed that the result of the propagation of electrons from the electron gun to the specimen is a plane wave. Partial coherence and/or convergent spherical illumination can then be accounted for by a partially coherent superposition of a set of plane waves. 

For the NFS spectrum of [Fe(tpa)(NCS)2] recorded at 108 K, which exhibits a HS to LS ratio of about 1 1, a coherent and an incoherent superposition of the forward scattered radiation from 50% LS and 50% HS isomers was compared, each characterized by its corresponding QB pattern (Fig. 9.16) [42]. The experimental spectrum correlates much better with a purely coherent superposition of LS and HS contributions. However, this observation does not yield the unequivocal conclusion that the superposition is purely coherent, because in the 0.5 mm thick sample the longitudinal coherence predominates since many HS and LS domains lie along the forward scattering pathway. In order to arrive at a more conclusive result, the NFS measurement ought to be performed with a smaller ratio aJD on a much thinner sample. Such an experiment would require a sample with 100% eiuiched Fe and a much higher beam intensity. 

From the point of view of the study of dynamics, the laser has three enormously important characteristics. Firstly, because of its potentially great time resolution, it can act as both the effector and the detector for dynamical processes on timescales as short as 10 - s. Secondly, due to its spectral resolution and brightness, the laser can be used to prepare large amounts of a selected quantum state of a molecule so that the chemical reactivity or other dynamical properties of that state may be studied. Finally, because of its coherence as a light source the laser may be used to create in an ensemble of molecules a coherent superposition of states wherein the phase relationships of the molecular and electronic motions are specified. The dynamics of the dephasing of the molecular ensemble may subsequently be determined. 

Ultrafast laser excitation gives excited systems prepared coherently, as a coherent superposition of states. The state wave function (aprobabihty wave) is a coherent sum of matter wave functions for each molecule excited. The exponential terms in the relevant time-dependent equation, the phase factors, define phase relationships between constituent wave functions in the summation. 

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

The photochemical excitation delivered by a narrowly defined pump laser pulse achieves three indispensable things it sets time = 0, energizes the reactant molecules, and localizes them in space. It induces molecular coherence as excitation of each of the individual molecules involved leads to a coherent superposition of separate wave packets, a highly locahzed, geometrically well-defined and moving packet—analogous to a classical system, one that can be described using classical concepts of atomic positions and momentum. 

This experimental work on the dissociation of excited Nal clearly demonstrated behavior one could describe with the vocabulary and concepts of classical motions.The incoherent ensemble of molecules just before photoexcitation with a femtosecond laser pump pulse was transformed through the excitation into a coherent superposition of states, a wave packet that evolved as though it represented a single vibrationally activated molecule. 

For nondegenerate superposed states the absorbers are in a coherent superposition of two nondegenerate excited states 1) = In ) and 2) = h pq). This superposition can be achieved in isotropic semiconductors for nf=n or by magnetic field... 

In our experiment, the coherent superposition of u = 0 and v=l state is created by the impulsive Raman transition. Due to the selection rule of a Raman transition. 

In the presence of the field, the molecular states are coherent superpositions of the states AM ) A ). In principle, the basis set must include an infinite number of states A ). However, the Floquet Hamiltonian matrix is block-diagonal and the diagonal matrix elements of the Floquet matrix separate in values SiS k-k increases. This suggests that it may be possible to include in the basis set a finite number of states from - max to max seek convergence with respect to In other words, the eigenstates of the Floquet Hamiltonian... 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

Next, we focus on the dynamics after ultrafast IR excitation of the main absorption peaks (cf. Fig. 1). The laser pulse spectra are shown in Fig. 2. Clearly, they are broad enough to excite a coherent superposition, i.e., a vibrational wave packet, in both cases. The dynamics is analyzed using the expectation values of the energy of the H/D reaction coordinates, Exy(t) = Ea a( I (t) a) a I (t)), and of the energies of the uncoupled skeleton modes,... 

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 theory of laser control of chemical reactions may be classified into two different domains Laser control by continuous wave (CW) lasers and by laser pulses. The former includes, for example, the strategies of (i) vibra-tionally mediated chemistry [1] and (ii) coherent superpositions of independent excitation routes [2] for experimental demonstrations see (i) Ref. 3 and... 

B. Kohler My question to T. Softley ties in to one of the major themes of the meeting, namely coherence. In your presentation you briefly mentioned that it may be important to consider an initially coherent superposition of states in the preparation step of experiments on highly excited Rydberg states. Several groups have now prepared coherent electronic wavepackets using picosecond (and shorter) pulses. Would this kind of an initial state be useful for any of the classes of pulsed-field ionization experiments that you have described ... 

T. P. Softley There is little doubt that in most ZEKE experiments using nanosecond lasers the Rydberg level structure is so dense that a coherent superposition of levels is populated initially, and the correct description of the dynamics should be a time-dependent one. It is possible that some control over the dynamics could be achieved using some of the methods described earlier in the conference, for example, simultaneous excitation through three-photon and one-photon transitions, using third-harmonic generation. 

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. 

When the fine structure frequencies fall below 100 MHz they can also be measured by quantum beat spectroscopy. The basic principle of quantum beat spectroscopy is straightforward. Using a polarized pulsed laser, a coherent superposition of the two fine structure states is excited in a time short compared to the inverse of the fine structure interval. After excitation, the wavefunctions of the two fine structure levels evolve at different rates due to their different energies. For example if the nd3/2 and nd5/2 mf = 3/2 states are coherently excited from the 3p3/2 state at time t = 0, the nd wavefunction at a later time t can be written as40... 

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