Stochastic baths

Before going on to consider more complicated systems, we review here some of the basic behavior of a two-state quantum system in the presence of a fast stochastic bath. This highly simplified bath model is useful because it allows qualitatively meaningful results to be obtained from a density matrix calculation when bath correlation functions are not available in fact, the bath coupling to any given system operator is reduced to a scalar. In the case of the two-level system, analytic results for the density matrix dynamics are easily obtained, and these provide an important reference point for discussing more complicated systems, both because it is often possible to isolate important parts of more complicated systems as effective two-level systems and because many aspects of the dynamics of multilevel systems appear already at this level. An earlier discussion of the two-level system can be found in Ref. 80. The more... 

Here, Qa and Fa are Hermite operators in the system and the stochastic bath subspaces, respectively. The deterministic system Liouvillian operators C t) and s are defined as... 

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

It is known that the interaction of the reactants with the medium plays an important role in the processes occurring in the condensed phase. This interaction may be separated into two parts (1) the interaction with the degrees of freedom of the medium which, together with the intramolecular degrees of freedom, represent the reactive modes of the system, and (2) the interaction between the reactive and nonreactive modes. The latter play the role of the thermal bath. The interaction with the thermal bath leads to the relaxation of the energy in the reaction system. Furthermore, as a result of this interaction, the motion along the reactive modes is a complicated function of time and, on average, has stochastic character. 

The brief review of the newest results in the theory of elementary chemical processes in the condensed phase given in this chapter shows that great progress has been achieved in this field during recent years, concerning the description of both the interaction of electrons with the polar medium and with the intramolecular vibrations and the interaction of the intramolecular vibrations and other reactive modes with each other and with the dissipative subsystem (thermal bath). The rapid development of the theory of the adiabatic reactions of the transfer of heavy particles with due account of the fluctuational character of the motion of the medium in the framework of both dynamic and stochastic approaches should be mentioned. The stochastic approach is described only briefly in this chapter. The number of papers in this field is so great that their detailed review would require a separate article. 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

Previously, stochastic Schrodinger equations for a quantum Brownian motion have been derived only for the particle component through approximated equations, such as the master equation obtained by the Markovian approximation [18]. In contrast, our stochastic Schrodinger equation is exact. Moreover, our stochastic equation includes both the particle and the field components, so it does not rely on integrating out the field bath modes. 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

It has been shown that for the case in which the pumping and probing lasers do not overlap, one can use the GLRT. In this section, it shall be shown how the GLRT can be applied to calculate the ultrafast time-resolved spectra. For this purpose, start from the stochastic Liouville equation to describe the EOM for the density matrix system embedded in a heat bath... 

Besides, the notation Q(f) for the time-dependent H-bond bridge coordinate interacting with the thermal bath may be viewed as the coordinate of a Brownian oscillator Q(f), which is a time-dependent stochastic variable ... 

The response function R2, which develops on the vibrational ground state after the second fight interaction, is the same as Ri [see Equation (7)]. This is a consequence of the stochastic ansatz in Equation (3), which implies that the bath influences the vibrational frequency of the solute but excitation of... 

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