Quantum dynamics oscillators

Figure 6.19 Quantum dynamics simulations for the two distinct situations of selective population of the (a) upper and (b) the lower target state. The frame (iv) shows the population dynamics induced by the shaped laser field pictured in (iii). The remaining panels depict (ii) the oscillations of the laser field together with the induced dipole moment and (1) the induced energetic splitting in the X-A-subsystem along with the accessibility of the target states. Gray backgrounds highlight the relevant time windows that are discussed in the text.

The parameter is the damping constant, and (n) is the mean number of reservoir photons. The quantum theory of damping assumes that the reservoir spectrum is flat, so the mean number of reservoir oscillators (n) = ( (O)bj(O j) = ( (1 / ) — 1) 1 in the yth mode is independent of j. Thus the reservoir oscillators form a thermal system. The case ( ) = 0 corresponds to vacuum fluctuations (zero-temperature heat bath). It is convenient to consider the quantum dynamics of the system (56)-(59) in the interaction picture. Then the master equation for the density operator p is given by... 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

We now briefly summarize the key results of the analysis of Refs. [50,51] for a reduced XT-CT model of the TFB F8BT heterojunction, using explicit quantum dynamical (MCTDH) calculations for a two-state model parametrized for 20-30 phonon modes. At this level of analysis, an ultrafast ( 200 fs) XT state decay is predicted, followed by coherent oscillations, see Fig. 8 (trace exact in panel (a)). Further analysis in terms of an effective-mode model and the associated HEP decomposition (see Sec. 4.2) highlights several aspects ... 

Thus, with this result for the kinetic energy and Eq. (E.10) for the potential energy, we conclude that the quantum dynamics of the normal modes is just the dynamics of n uncoupled harmonic oscillators that is,... 

Gray, S.K. (1987) A periodically forced oscillator model of van der Waals fiagmentation Classical and quantum dynamics, J.Cfien/.Pfivs- 87, 2051-2061. 

Problems involving harmonic oscillators that are shifted in their equilibrium positions relative to some preset origin are ubiquitous in simple models of quantum dynamical processes. We consider a few examples in this section. 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

The prominence of these quantum dynamical models is also exemplified by the abundance of theoretical pictures based on the spin-boson model—a two (more generally a few) level system coupled to one or many harmonic oscillators. Simple examples are an atom (well characterized at room temperature by its ground and first excited states, that is, a two-level system) interacting with the radiation field (a collection of harmonic modes) or an electron spin interacting with the phonon modes of a surrounding lattice, however this model has found many other applications in a variety of physical and chemical phenomena (and their extensions into the biological world) such as atoms and molecules interacting with the radiation field, polaron formation and dynamics in condensed environments. 

The Hamiltonian in Eq. (4.1) has an almost product-like form since the majority of coordinates are treated as harmonic oscillators. This makes it rather suitable for quantum dynamics simulations, either in the time-dependent Hartree approximation [31] or using the more general multi-configuration time-dependent Hartree approach [36, 37]. 

To test the CMD approach on a system for which numerically exact quantum dynamical results can be obtained, a one-dimensional nonlinear oscillator model was employed, given by [4, 5]... 

Figure 4 shows two typical trajectories which trace the quantum mechanical wave-packet motion. They are started on the symmetric stretch line with different initial momenta pointing into the exit channels. The two trajectories represent dissociation into H + OD and D + OH, respectively. What was found for the quantum dynamics is more clearly demonstrated by the classical trjectories the H + OD dissociation is faster and the oscillations of both trajectories around the minimum energy path clearly shows the vibrational excitation of the fragments. Indeed, if we compute the time-evolution of the bondlength expectation values th d) from the bifurcated packets it is found that they resemble closely the classical trajectories (as can be expected from Ehrenfest s theorem). The wave-packet motion shows that the dissociation proceeds as can be anticipated classically. 

Quantum dynamical calculations of the pump probe spectra for the two isotopes were performed for delay times up to 40 ps. A comparison of the experimental and theoretical ionization signals as a function of the delay time is presented in Fig. 3.15. In agreement with the experimental data, the short-time dynamics of the theoretical signal show the 500 fs oscillation period of the wave packet prepared in the A state (centered around v = 11) and the long time dynamics reflect the totally different beat structures of the two isotopes. However, the oscillation periods of the pronounced regular beat structure of the isotope (Fig. 3.15 a) and of the weak, irregular... 

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