Non-uniraty evolutios and non hermitina hamiltonians in decoherence of closed systems

M. CASTAGNINO; S. FORTIN

Paris

Congreso; Non-uniraty evolutios and non hermitina hamiltonians in decoherence of closed systems; 2012

Paris Diderot University

Non-unitary evolutions are essential to explain and study the phenomena of decoherence, the quantum to the classical limit, and the 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, and non-Hermitic Hamiltonians.As it is indicated in the brief historical summary of paper [6], schematically three periods can be identified in the development of the general program of decoherence:?    In the first period the arrival to the equilibrium of irreversible systems was studied. The main problem of this period was that too long decoherence times were found, if compared with the experimental ones. ?    In a second period decoherence in open systems was studied. The main model for this process was EID that determines, case by case, which is the privileged basis, called usually moving preferred basis where decoherence takes place in a decoherence time  . This is the orthodox position on the subject [7]. In this period, the decoherence times founded were much smaller, solving the problem of the first period.?    Recently, in a third period, the study of the arrival to equilibrium of closed systems was studied. Within this period, many formalisms for closed systems were introduced (see [8]). Among them a new approach to decoherence, SID, was presented by Castagnino et al., endowed with a non-Hermitian Hamiltonian and therefore with a non unitary evolution. According to this approach decoherence is a process dependent of the choice of some observables which have a particular physical relevance (the van Hove observables) in a closed system. This process also determines which is the privileged basis, called final preferred basis that defines the observables that acquires classic characteristics (at the relaxation time ) and that can be interpreted like properties that obey a Boolean logic. SID also solves an important problem explaining how closed system reach equilibrium. Relaxation time can be very long but equilibrium is always reached. In fact, Riemann-Lebesgue theorem shows that, in any system with destructive interference, reaches equilibrium in a very long Khalfin: time [9]. This time was experimentally detected in 2006 [10]. Moreover, in this contribution (following [2]) , we will introduce a generic definition of moving preferred basis where the state decoheres in a time  , also solving the main problem of the first period.In papers [1] we have introduced a proposal of a general definition of moving preferred basis for open systems, using of the usual formalism for these systems known as Environment Induced Decoherence (EID) .In this contribution we will extend our definition to closed system for a particular formalism known as Self Induced Decoherence [2]. 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 [2]).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 the decoherence time [1].Then, the evolution of the system can be decomposed in three periods.i.- When the time is smaller that decoherence time the closed 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 closed 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.We will prove that our definitions coincide with the one of Roland Omnès [3] and  we discuss a similar analogy for closed systems: the Casati-Prosen model [4]. Our formalism is essentially based in papers [5].So in this work we focused the attention on the closed systems approach. Our main aim is to present a new conceptual perspective that will clarify some points that still remain rather obscure in the literature on the subject, e. g. the possibility of decoherence in closed systems  and the definition of the moving preferred basis in these systems.