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arXiv:1304.3190 · Journal of Physics A Mathematical and Theoretical

Non-Hermitian Hamiltonians in decoherence and equilibrium theory

Mario Castagnino, Sebastian Fortin

Published Oct 23, 2012• 15 citations
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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. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.

DOI
10.1088/1751-8113/45/44/444009

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Journal of Physics A Mathematical and Theoretical•Oct 23, 2012•53 citations
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 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 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.
The main motivation behind the call for this special issue was to gather recent results, developments and open problems in quantum physics with non-Hermitian operators. There have been previous special issues in this journal [1, 2] and elsewhere on this subject. The intention of this issue is to reflect the current state of this rapidly-developing field. It has therefore been open to all contributions containing new results on non-Hermitian theories that are explicitly -symmetric and/or pseudo-Hermitian or quasi-Hermitian. In the last decade these types of systems have proved to be viable self-consistent physical theories with well defined unitary time-evolution and real spectra. As the large number of responses demonstrates, this is a rapidly evolving field of research. A consensus has been reached regarding most of the fundamental problems, and the general ideas and techniques are now readily being employed in many areas of physics. Nonetheless, this issue still contains some treatments of a more general nature regarding the spectral analysis of these models, in particular, the physics of the exceptional points, the breaking of the -symmetry, an interpretation of negative energies and the consistent implementation of the WKB analysis. This issue also contains a treatment of a scattering theory associated with these types of systems, weak measurements, coherent states, decoherence, unbounded metric operators and the inclusion of domain issues to obtain well defined self-adjoint theories. Contributions in the form of applications of the general ideas include: studies of classical shock-waves and tunnelling, supersymmetric models, spin chain models, models with ring structure, random matrix models, the Pauli equation, the nonlinear Schrödinger equation, quasi-exactly solvable models, integrable models such as the Calogero model, Bose–Einstein condensates, thermodynamics, nonlinear oligomers, quantum catastrophes, the Landau–Zener problem and pseudo-Fermions. Applications close to experimental realization are proposed in optics, including short light pulse models, waveguides and laser systems, and also in electronics. We hope that this issue will become a valuable reference and inspiration for the broader scientific community working in mathematical and theoretical physics. References [1] Fring A, Jones H F and Znojil M (ed) 2008 J. Phys. A: Math. Theor. 41 240301 [2] Geyer H, Heiss D and Znojil M (ed) 2006 J. Phys. A: Math. Gen. 39 9963