Explanation of decoherence and quasi-equilibrium in systems with few degrees of freedom
demands a deep theoretical analysis that considers the observed system as an open quantum system. In this work, we study the problem of decoherence of an observed system of quantum interacting particles coupled to a quantum lattice. Our strategy is based on treating the environment and the system-environment Hamiltonians fully quantum mechanically, which yields a representation of the time evolution operator useful for disentangling the different time scales underlying the observed system dynamics. This general approach is applied to study reversible decoherence in protonnuclear magnetic resonance (1 H NMR) of nematic liquid crystals. A summary of this work is as follows. We assume that the spin evolution can be divided into three main stages: reversible decoherence (closed dynamics), irreversible decoherence and relaxation. Focusing only on the reversible process, we calculate the decoherence function and apply it to describe the evolution of a coherent spin state, induced by the coupling with the molecular environment, in the absence of spin- lattice relaxation. The molecular environment is assumed to follow a mean field dynamics (Maier- Sauper theory [1]). A process based on the study of the irreducible representation of the Lie group E(2) (euclidean group) [2], allows us to quantize the Maier-Saupe potential and thus to describe the
molecular environment quantum mechanically as well as the spin-lattice interaction. To our
knowledge, this is the first time the quantization scheme presented in [2] has been used to quantize the molecular and the spin-lattice interactions.
Contrary to previous works on the subject [3], it is found that a mean-field theory would satisfy all the requirements for introducing an irreversible decoherence. However, when the model is applied to obtain the decoherence function in the study of the NMR free induction decay (FID), we find that the theoretical decay is almost twice as fast as the experimental one [3]. The sources of discrepancy could be the following: (1) a mean-field theory is an oversimplification of the true molecular dynamics and thus not suitable for describing the observed phenomena. (2) the assumption of a time scale where reversible decoherence takes place may not be possible, suggesting that the entire spin dynamics is irreversible.

[1] Stephen, M. J., & Straley, J. P. (1974). Physics of liquid crystals. Rev. Mod. Phys., 46(4), 622.
[2] Kastrup, H. A. (2006). Quantization of the canonically conjugate pair angle and orbital angular
momentum. Phys. Rev. A, 73(5).
[3] Segnorile, H. H., & Zamar, R. C. (2011). Quantum decoherence and quasi-equilibrium in open
quantum systems with few degrees of freedom: Application to 1H NMR of nematic liquid crystals.
The Journal of Chemical Physics, 135(24).