Population decay

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

It has been demonstrated that the whole photoexcitation dynamics in m-LPPP can be described considering the role of ASE in the population depletion process [33], Due to the collective stimulated emission associated with the propagation of spontaneous PL through the excited material, the exciton population decays faster than the natural lifetime, while the electronic structure of the photoexcited material remains unchanged. Based on the observation that time-integrated PL indicates the presence of ASE while SE decay corresponds to population dynamics, a numerical simulation was used to obtain a correlation of SE and PL at different excitation densities and to support the ASE model [33]. The excited state population N(R.i) at position R and time / within the photoexcited material is worked out based on the following equation ... 

The previous sections focused on the case of isolated atoms or molecules, where coherence is fully maintained on relevant time scales, corresponding to molecular beam experiments. Here we proceed to extend the discussion to dense environments, where both population decay and pure dephasing [77] arise from interaction of a subsystem with a dissipative environment. Our interest is in the information content of the channel phase. It is relevant to note, however, that whereas the controllability of isolated molecules is both remarkable [24, 25, 27] and well understood [26], much less is known about the controllability of systems where dissipation is significant [78]. Although this question is not the thrust of the present chapter, this section bears implications to the problem of coherent control in the presence of dissipation, inasmuch as the channel phase serves as a sensitive measure of the extent of decoherence. 

Fig. 7. Thermally populated / -decay channels from "Tc to "Ru [38], Relevant proton(n) and neutron(v) shells are shown as compartments filled with nucleons (x)...

In the secular approximation [89], we can eliminate the coherence terms [e.g., pr, (x)(u / u")] in Eq. (III.9) such that the only diagonal terms contribute to the vibrational transitions through which the vibrational populations in various states are coupled. By applying the ladder model [89] to the interaction between the vibrational and heat-bath modes, the vibrational population decay constant is expressed as... 

By omitting the pure dephasing processes, which is warranted at low temperatures, the dephasing constant 1) ), in Eq. (III. 19) can be expressed, in terms of the population decay constants of the states v and v , as... 

The Monod equation differs from the Michaelis-Menten equation in that it includes as a factor biomass concentration [X], which can change with time. A microbe as it catalyzes a redox reaction harvests some of the energy liberated, which it uses to grow and reproduce, increasing [X], At the same time, some microbes in the population decay or are lost to predation. The time rate of change in biomass... 

To summarize, the reactive flux method is a great help but it is predicated on a time scale separation, which results from the fact that the reaction time (1/T) is very long compared to all other times. This time scale separation is valid, only if the reduced barrier height is large. In this limit, the reactive flux method, the population decay method and the lowest nonzero eigenvalue of the Fokker-Planck equation all give the same result up to exponentially small corrections of the order of For small reduced barriers, there may be noticeable differences between the different definitions and as aheady mentioned each case must be handled with care. 

This coupling means that there is no population decay to the bath and the decoherence is only via the phase. In other pioneering works [39-41], off-diagonal coupling to a Lorentzian bath was considered within the RWA. The Hamiltonians in all of these works should be contrasted with the more general Hamiltonians (4.18), which account for both dephasing and decay into arbitrary baths, without the RWA. 

If the system under consideration possesses non-adiabatic electronic couplings within the excited-state vibronic manifold, the latter approach no longer is applicable. Recently, we have developed a simple model which allows for the explicit calculation of RF s for electronically nonadiabatic systems coupled to a heat bath [2]. The model is based on a phenomenological dissipation ansatz which describes the major bath-induced relaxation processes excited-state population decay, optical dephasing, and vibrational relaxation. The model has been applied for the calculation of the time and frequency gated spontaneous emission spectra for model nonadiabatic electron-transfer systems. The predictions of the model have been tested against more accurate calculations performed within the Redfield formalism [2]. It is natural, therefore, to extend this... 

Here Tr means the trace over vibrational (not electronic) coordinates, pf = Z x exp(—ha/kT), and Za are the corresponding partition functions. The proposed damping operator consists of the two contributions, re and which are responsible for the electronic and vibrational relaxation, respectively. The first term in the operator (2) reflects the excited-state population decay, so that T) = l/ e is simply the excited... 

Figure 6. Population decay of the initial state in a barrierless double-well system calculated using multilevel Redfield theory [25]. The vibrational frequency is 60 cm1. (----------)...

Since his arrival at McMaster in 1988, Randall Dumont has focused on statistical theories and their origin in quantum and classical mechanics. His interests include the development of Monte Carlo implementations of statistical theory wherein dynamical processes are simulated by random walks on potential energy surfaces. The breakdown of statistical theory and the appearance of nonexponential population decay are also topics of his ongoing investigations. Other questions of interest are the incorporation of quantum effects into statistical theory and the effects of collisions on reaction processes. He has a special interest in argon cluster evaporation in vacuum197 and in the description of simple isomerization reactions.198 His other interests include the semiclassical description of classically unallowed processes such as tunneling.199... 

We have so far been able to obtain exact explicit analytic solutions for (a) the case where only processes (i) and (ii) are significant, and (b) the case where only processes (ii) and (iii) are significant. We have also obtained an approximate analytic solution for the case where all three processes (i), (ii) and (iii) occur, but where the loss of radicals occurs predominantly by process (ii) rather than by prodess (iii). As a generalisation of case (a), we have obtained a general solution which covers the case where the parameters which characterise the processes (i) and (ii) are themselves time-dependent. The general solution to case (b) requires modification if processes of type (ii) do not occur. Complete solutions have been obtained for three special cases of (b), namely, decay from a Stockmayer-01Toole distribution of locus populations, decay from a Poisson distribution of locus populations, and decay from a homogeneous distribution of locus populations. 

Various models are used in the literature to account for the kinetics of the excitons involved in optical processes. In the simplest cases, the signal evolution n(t) can be reproduced by considering either a single exponential or multiexponential time dependences. This model is well suited for solutions or solids in which monomolecular mechanisms happen alone. Since in most transient experiments the temporal response is a convolution of a Gaussian-shaped pulse and of the intrinsic kinetics, the rate of change with time of the excited-state population decaying exponentially is given by... 

Employing V — b)F bQ exp(— J2j xj9j) b as the system-heat bath interaction, where qj denotes the vibrational coordinate of theyth heat bath mode, the vibrational population decay rate constant is obtained as... 

Large nonlinearities based on saturated absorption or bandfilling effects are reported for semiconductors. The response of these nonlinearities is fast but recovers only slowly due to the created excited state population. Decay times of the excited states on the order of some hundred picoseconds to nanoseconds are detrimental for all-optical switching with large repetition rates. 

Si(7171 ) absorption band of these molecules [167, 168]. The experimental scheme is depicted in Fig. 19, showing energetics for the case of OHBA. Excitation with a tuneable pump laser /zvpump forms the enol tautomer in the Si (7171 ) state. The ESIPT leads to ultrafast population transfer from the Si enol to the Si keto tautomer. On a longer time scale, the Si keto population decays via internal conversion to the ground state. Both the enol and keto excited-state... 

All femtosecond-resolved transients in Fig. 3a can be best fitted by a sum of a series of exponential functions. These functions can be separated into two parts One part represents solvation processes, and the other one is for lifetime emissions (population decay). The transient signal can be written as follows ... 

Nonadiabatic ab initio surface hopping simulations were carried out for 11 different starting points taken from a ground state run at 100 K. The excited state nonradiative lifetime has been determined to be 1.7 ps from an exponential fit to the 5) population decay (Eq. (10-11)) and to lie in the interval [1.7.. .3.1.. .13.4] ps using the average transition probability (Eq. (10-13)). 

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