Non-unitary evolutions and non-hermitian hamiltonians in decoherence and equilibrium tehory

SEBASTIAN FORTIN; MARIO CASTAGNINO

Dresden

Seminario; International Seminar: Quantum Physics with Non-Hermitian Operators; 2011

Max Planck Institute of Physics

Non-unitary evolutions are essential to explain and study the phenomena of decoherence, the quantum to classical limit, and final equilibrium. These phenomena appear in the evolution of quantum system, where decoherence time and relaxation time can be defined using non-unitary evolutions, pole theory (1), and non-Hermitic Hamiltonians.  Decoherence time is always smaller than relaxation time. After decoherence time the state of the system becomes diagonal in a ?moving preferred basis? that must be defined. After relaxation time the system reaches equilibrium.Moreover, there are many formalisms to describe quantum decoherence, however, many of them give a non general and  ad hoc definition of "pointer basis" or "moving preferred basis", and this fact is a problem for the decoherence program. In this lecture (following paper (2)) we will consider quantum systems under a general theoretical framework for decoherence and present a tentative very general definition of the moving preferred basis, which is implemented in a well known open system model (3). The obtained decoherence and the relaxation times are defined and compared with those of this model.As usual we will consider a closed system U and we will define two subsystems: S, the ?proper or open system?, and E, the environment. Then we define the generic observable of the system as the tensor product of an observable of the proper system and the unit operator of the environment. It is well known that in this case the state of the proper system is the partial trace of the states of the system, where the environment has being ?traced away?.Then, if we consider the Hermitic Hamiltonian H of U and the inner product of the evolved state of the proper system and any observable of the proper system we can make its analytical continuation, in the energy variable into the lower complex half-plane, and in general we will find  poles. These poles define  all the possible non-unitary decaying modes with characteristic decaying times proportional to the inverse of the imaginary part of the poles (we do not consider the Khalfin mode since it has extremely long decaying time (1)).From these characteristic times we can deduce the relaxation time, which turns out to be the inverse of the imaginary part of the pole closest to the real axis and therefore it is the largest characteristic time. We can also deduce  the decoherence time, that turns out to be a function of the imaginary part of the poles and the initial conditions of the system. Moreover with the same elements a moving preferred basis can be defined, where the state of the system becomes diagonal at decoherence time (2).Then the evolution of the system can be decomposed in three periods.i.- When the time is smaller that decoherence time the open system has a pure quantum evolution according to a quantum statistic. The non Hermit Hamiltonian corresponding to this non-unitary evolution can be defined.   ii.- When the time is greater than the decoherence time and smaller than the relaxation time the state of the open system becomes diagonal in the moving preferred basis and it evolves according to a classical statistic (then actualization or collapse may occur) iii.- When the time is greater than the relaxation time the system is classical (precisely if collapsed or actualized it evolves according to a classical mechanics) and finally it reaches equilibrium.The detailed explanation of these three phases completes the quantum to classical limit.All these results are compared with those of a well known Omnès? model (3) with good results. For this comparison the non Hermitic Hamiltonian is essential. Moreover, the discussion above corresponds to an Environment Induced Decoherence (EID) model, but we are almost sure that it can be extended to other kinds of models, e. g. those of papers (4).Our main contribution is that we have introduced general definitions for the decoherence time for the relaxation times and for the moving preferred basis, that were defined only case by case in the literature (5)