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Journal of Molecular Structure

Optical bloch equations: disagreement between experiment and theory explained by quantum dynamical semigroups

K. Lendi

Published Mar 1, 1986• 2 citations
Publisher
DOI
10.1016/0022-2860(86)85126-2

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The well-known magnetic or optical Bloch equations are extended to the case of unstable systems such as, e.g., spontaneously decaying particles and molecules undergoing chemical reactions or decaying from excited states back to states other than those involved in resonance. The modified equations are derived from a general theory of irreversible processes where quantum Markovian master equations and the corresponding completely positive dynamical semigroups dictate the time evolution of density operators. The results are of particular importance in those situations where decay occurs level-selectively and all relaxation and decay constants are of comparable order of magnitude. The obtained generalized Bloch equations are of direct applicability to problems of time-resolved magnetic or optical resonance spectroscopy.
Various aspects of a rather general treatment of proton-spin relaxation in water are discussed within the framework of quantum-dynamical semigroup theory for a four-level system coupled to a reservoir in equilibrium. In particular, the specifications of the infinitesimal generator of time evolution, either in Kossakowski or in Davies form, are worked out in detail. With the help of the Lie algebra of SU(4), the results are used to derive, under suitable simplifications, generalized Bloch equations for the static and alternating-field case. The relevant correlation functions are calculated using conventional approaches but supplemented by taking into account explicitly results from a stochastic model for formation and breaking of hydrogen bridges. A further approximate reduction of the coupled general equations to simpler ordinary Bloch equations leads to an identification of the relevant relaxation times. This approach provides a somewhat different interpretation of rotational correlation times whose numerical values are estimated over a wide temperature range.
In a recent experiment [ DevoeR. G.BrewerR. G., Phys. Rev. Lett.50, 1269 ( 1983)], it was found that the optical Bloch equations could not satisfactorily explain the signal that was observed for free-induction decay in the impurity ion cyrstal Pr3+:LaF3. Several theories have been proposed to explain this failure of the Bloch equations. In this paper, the general validity conditions for the optical Bloch equations are examined within the limits of a Markovian relaxation model. The specific problem to be considered is the interaction of an optical field with two-level atoms. The atoms undergo relaxation as a result of coupling to a perturber bath that, itself, is negligibly affected by the relaxation process. First, a simple decay-parameter model is assumed for the relaxation of atomic density-matrix elements. Such a model is found to lead to a set of generalized Bloch equations of which the conventional Bloch equations form a subset. Subsequently, more-realistic models for relaxation in both vapors and solids are considered within the limits of the impact approximation (i.e., the duration of a fluctuation can be viewed as instantaneous with respect to all relevant time scales in the problem). It is found that, even in the impact (Markovian) approximation, the generalized Bloch equations cannot be expected to provide an adequate description of relaxation, owing to effects in which a fluctuation-induced change in the atomic transition frequency persists between fluctuations. In vapors this frequency shift is produced by velocity-changing collisions that change the atomic resonance frequency (as seen in the laboratory frame), whereas in solids it is produced by local-field fluctuations. The limiting conditions under which one can expect both the generalized and the conventional Bloch equations to retain their validity are explored.