Abstract
The process of vibrational relaxation was studied in this chapter, specifically from the point of view of deriving generalizations of the Montroll-Shuler equation of the process into the coherent domain. The process of vibrational relaxation of molecules is interesting for various reasons including its interaction with luminescence and for its analysis as a gateway process for more drastic processes of dissociation or chemical reactions. Coarse-grained projection operators were developed to eliminate reservoir variables and focus only on the relaxation of the vibrational coordinate of the molecule under consideration. Several models of the interaction of a vibrationally relaxing molecule embedded in a reservoir were studied. Some surprising effects were described for the case of strong coupling or multiphonon processes. This chapter should be useful both to understand the (coarse-grained) projection technique in microscopically specified reservoirs and to probe the process of the approach to equilibrium of systems in contact with a bath, including its temperature dependence. The latter is of fundamental importance to the general subject of quantum non-equilibrium statistical mechanics.
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Notes
- 1.
The calculated quantities are, nevertheless, all observable in principle.
- 2.
A student at an oral exam in Rochester in the early 80s had kindly referred to my own work with the Montroll-Shuler equation in his writeup. I was one of the examiners. Another examiner on the committee attacked that equation and the rest of us with it because it was, after all, a Master equation and consequently incapable of analyzing very short-time behavior. On hearing that objection, I had quietly resolved to generalize the Montroll-Shuler equation as soon as the opportunity arose. It only came to pass, almost four decades later, during conversations with two of my last students, Matt Chase and Anastasia Ierides, and is now presented to you in this chapter. I must confess it certainly is not yet capable of addressing questions that depend on off-diagonal density matrix elements directly, or on initial conditions that are not random phase. Incidentally, the student was Wayne Knox, the son of my collaborator Bob Knox, and an illustrious scientist in his own right who went on to direct the Institute of Optics at Rochester as also the Advanced Photonics Research at Bell Labs.
- 3.
Montroll and Shuler had been successful in achieving great simplifications in the analysis of their equation on the basis of a property known as canonical invariance. For instance, it allowed one to show that an initial probability distribution of the Boltzmann form among the molecular energy levels at a temperature T other than that of the environment, maintains that Boltzmann form throughout its evolution from the initial T to the environment T. This permits the introduction of the concept of a time-dependent T in the evolution of the probabilities. This basic property, also explored further in Andersen et al. (1964), van Kampen (1971), and Thiele et al. (1981), is unfortunately lost in going from the Master equation to the GME. Given the interesting phenomena that Seshadri and I had been able to describe by exploiting canonical invariance in our extensions of the equation to include additional processes (Seshadri and Kenkre 1976; Kenkre and Seshadri 1977; Kenkre 1977b), we wanted to be able to restore it to the GME in some modified form. Chase and I made multiple attempts in this direction. So far they have not met with success.
- 4.
It should be evident that we are not offering any new Hamiltonians and methods to deal with them in what follows. The techniques of solution are borrowed from earlier work of many authors in the excitonic polaron field, most notably Silbey (we have given examples and explanations in previous chapters). What is new is the consequences for the present vibrational relaxation problem, in particular a comparison of results for strong versus weak coupling, as we shall see below.
- 5.
How this situation is connected to some recent discussions of coherence among photosynthesis scientists will be remarked on in Chap. 16.
- 6.
For instance, do the two happen sequentially? Simultaneously? What in the system or bath decides the order? The non-resonant oscillations at short times in an inequivalent dimer are independent of the sign of the energy difference. The populations at long times are crucially dependent on the sign. How does the sensitivity to sign come about as time proceeds?
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Kenkre, V.M.(. (2021). Projections and Memories for Microscopic Treatment of Vibrational Relaxation. In: Memory Functions, Projection Operators, and the Defect Technique. Lecture Notes in Physics, vol 982. Springer, Cham. https://doi.org/10.1007/978-3-030-68667-3_7
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