Weakly coupled subsystems

We assume that V, the operator that couples systems L and R to each other, mixes only / and r states, that is, = Vr,r — 0- We are interested in the transition between these two subsystems, induced by V. We assume that (1) the coupling V is weak coupling in a sense explained below, and (2) the relaxation process that brings each subsystem by itself (in the absence of the other) into thermal equilibrium is much faster that the transition induced by V between them. Note that assumption (2), which implies a separation of timescales between the L 7 transition and the thermal relaxation within the L and R subsystems, is consistent with assumption (1). 

Environments with stable energetic stresses are frequently divided into nearly decoupled spatial or compositional subsystems. Tliis is true of quasi-stable energetic redox couples at hydrothermal vents and of tlie weakly coupled 6000 K spectrum of solar visible light and 300 K terrestrial tliemial black body [55]. Tlie separate components may constitute internally near-equilibrium subsystems, defined individually by simple ensemble constraints. 

The existence of statistical correlations between the subsystems leads to an overall situation less random than when the subsystems are statistically independent of (or very weakly coupled to) each other. 

Before we introduce the concept of an open system, it is useful to discuss the specific heat of the subsystem itself. The Hamiltonian "Xg in Equation 11.7 does qualify for describing the prototype thermodynamics of a smah quantum system, like a harmonic oscillator. What we have to do is to imagine Jig to be weakly coupled to a classical heat bath, with which the system undergoes exchange of energy. The consequent energy fluctuations provide the temperature of the system. All this can be put into statistical mechanical perspective in terms of the Gibbsian partition function... 

Special realizations of vibronic coupling systems, such as occur for well-localized, weakly coupled molecular subsystems have been touched upon only briefly, in Sect. 6.1.2.5. Another special case, that of light-induced Coins [99,100] has not been discussed above, but is expected to emerge as highly relevant in strong laser fields in the future. This may, of course, also apply to other realizations of specific sysfems, and likewise regarding their experimental detection. 

The Redfield theory is based on a weak coupling assumption and has been used extensively in nuclear magnetic resonance (NMR) theory for treating the influence of the surroundings on the spectra. Attempts to go beyond the weak coupling limit assumed in the Redfield theory have been made [78], and the requirement of complete positiveness assumed by Lindblad in the coupling between the subsystem and the reservoir has been challenged by Pechukas [79]. 

We now connect the analysis given above with the equation of motion displayed in Eq. (5.5). That equation of motion follows from subdivision of a system into an open subsystem S and a complementary reservoir R. When the coupling between S and R is weak, the evolution of the open system 5, due to the internal dynamics of 5 and the interaction with the reservoir R, can be described in density matrix form by Eq. (5.5). Now writing... 

Physically, we consider the parametric regime where coupling is so weak that it hardly induces any transition during At and, consequently, the effective Hamiltonian has a nondemolition form in the sense defined in the previous section. Since the interaction term commutes with both system Hamiltonians, the (expectation values of the) subsystem energies are constants of time. This is, in fact, a consequence of the assumed lack of resonances between the oscillator and the spin. 

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