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Non-Hermitian Hamiltonians in decoherence and equilibrium theory

Mario Castagnino, Sebastián Fortín

Published Oct 1, 2012• 11 citations
OpenAlex

Abstract

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 paper we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee- Friedrichs model.

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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 paper we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee- Friedrichs model.
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 paper, we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee–Friedrichs model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
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 paper 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. The obtained decoherence and the relaxation times are defined and compared with those of this model.
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 paper we will consider quantum systems under a general theoretical framework for decoherence and present a very general definition of the moving preferred basis. In addition, this definition is implemented in a well known model and the time of decoherence and the relaxation time are defined and compared with those of this model.
One of the challenges of the Environment-Induced Decoherence (EID) approach is to provide a simple general definition of the moving pointer basis or moving preferred basis. In this paper we prove that the study of the poles that produce the decaying modes in non-unitary evolution, could yield a general definition of the relaxation, the decoherence times, and the moving preferred basis. These are probably the most important concepts in the theory of decoherence, one of the most relevant chapters of theoretical (and also practical) quantum mechanics. As an example we solved the Omnès (or Lee–Friedrich) model using our theory.
We present a general analytic method for evaluating the generally time-dependent pointer states of a subsystem, which are defined by their capability not to entangle with the states of another subsystem. We explore the conditions under which the pointer states of the system become independent of time; so that a preferred basis of measurement can be realized. We relate the mathematical conditions for having time-independent pointer states to some classes of possible symmetries in the Hamiltonian of the total composite system. Indeed, our theory would serve as a generalization of the existing theory for determination of the preferred basis of measurement. By exploiting this new theory we can obtain those regimes of the parameter space for a given total Hamiltonian defining our system-environment model for which a preferred basis of measurement can be realized. Moreover, we can predict the corresponding preferred basis of measurement for each regime. We can also obtain the time-dependent pointer states of the system and the environment in most of the other regimes where the pointer states of the system are time-dependent and a preferred basis of measurement cannot be realized at all. This ability to obtain time-dependent pointer states is specifically important in decoherence studies; as these pointer states, although they evolve with time and cannot represent the preferred basis of measurement, they correspond to those initial conditions for the state of the system and the environment for which we can have longer decoherence times. In the next step, we consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system and the interaction with the environment do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we calculate the time-dependent pointer states of the system and the environment for the case that the environment initially is prepared in the coherent state. We also obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We will show that for the case that the system initially is prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will be a sinusoidal function with a slow decaying envelop which is characterized by a decay time proportional to n-bar.
We present a complete review of the quantum-to-classical limit of open systems by means of the theory of decoherence and the use of the Weyl-Wigner-Moyal (WWM) transformation. We show that the analytical extension of the Hamiltonian provides a set of poles that can be used to (a) explain the non-unitary evolution of the relevant system and (b) completely define the set of preferred states that constitute the mixture into which the system decoheres: the Moving Preferred Basis. Moreover, we show that the WWM of these states are the best candidates to obtain the trajectories in the classical phase-space.