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Coupling of Quantum Systems to the Environment: Functional Differential Treatment

Edward B. Manoukian, S. Sukkhasena

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

A functional differential treatment, via the quantum dynamical prin-ciple, is given for the coupling of quantum systems to the environment. As one is involved in taking the trace over the dynamical variables of the environment, the analysis necessarily deals with transition probabilities rather than with amplitudes. It is shown that the functional differential treatment is quite suitable for such a study as it involves in carrying out functional differentiations, with respect to classical sources, on func-tionals describing decoupled physical systems from the environment.

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A functional differential treatment, via the quantum dynamical prin-ciple, is given for the coupling of quantum systems to the environment. As one is involved in taking the trace over the dynamical variables of the environment, the analysis necessarily deals with transition probabilities rather than with amplitudes. It is shown that the functional differential treatment is quite suitable for such a study as it involves in carrying out functional differentiations, with respect to classical sources, on func-tionals describing decoupled physical systems from the environment.
Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum-classical sys- tems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approxi- mation. The dynamics of these composite systems are governed by a quantum-classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of tra- jectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many- body environments.
Quantum decoherence in all its locus of essence, and expression is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. Detrimental ramifications, and pernicious implications of this dephasing leads to classical or probabilistically additive behavior. Quantum decoherence gives the appearance of wave function collapse (the reduction of the physical possibilities into a single possibility as seen by an observer. Here it is to be noted that perception is not the reality and what you see is not what you see ;what you do not see is what you do not see; what you see is what you do not see ;and what you do not see is what you see.)… Thus in all its wide ranging manifestations, it justifies the propositional subsistence and corporeal reality that justifies the framework and intuition of classical physics as an acceptable approximation: decoherence is the mechanism by which the classical limit emerges out of a perceptual field   of quantum starting point and it determines the location of the quantum-classical boundary. Decoherence occurs when a system interacts with its environment in a thermodynamically irreversible way. This prevents transitive states, substantive sub states and determinate orientation different elements in the quantum superposition of the system and environment's wavefunction from interfering with each other. Decoherence has been a subject of active research since the 1980s. Decoherence can be viewed as the principal frontier of diurnal dynamics that results in loss of information from a system into the environment (often modeled as a heat bath), since every system is loosely coupled with the energetic state of its surroundings, with particularistic predicational pronouncements. . Viewed in isolation, the system's dynamics are non-unitary (although the combined system plus environment evolves in a unitary fashion). Thus, the dynamics of the system aphorism and anecdote of the system alone are irreversible. As with any coupling, entanglements are generated in its theme and potentialities between the system and environment, which have the effect of sharing quantum information with—or transferring it to—the surroundings. It is a blatant and flagrant misconception that collapse of wave function is attributable and ascribable to wave function collapse; Decoherence does not generate actual wave function collapse. It only provides an explanation for the appearance of the wavefunction collapse, as the quantum nature of the system leaks into the environment. So, the wave function collapse is the figment of the observer’s imagination, product of puerile prognostication and resultant orientationality of his phantasmagoria.  One cannot have the apodictic knowledge of reality that is; components of the wavefunction are decoupled from a coherent system, and acquire phases from their immediate surroundings. A total superposition of the global or universal wavefunction still exists (and remains coherent at the global level), but its ultimate fate remains an interpretational issue. This is a very important pharisaical provenience and plagenetious precocity with all the disembodied resemblances of the system in total form; specifically, decoherence does not attempt to explain the measurement problem. Rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive. Moreover, our observation tells us that this mixture looks like a proper quantum ensemble in a measurement situation, as we observe that measurements lead to the of precisely one state in the ensemble.Decoherence represents a challenge for the practical realization of quantum computers, since they are expected to rely heavily on the undisturbed evolution of quantum coherences. Simply put; they require that coherent states be preserved and that decoherence is managed, in order to actually perform quantum computation. In the following we give a model for decoherence of the environment, coupled quantum states, at determinate and differential levels ,internal resonance in coupled spin systems, and the conflict in the Quantum state operations and decoupling. KEY WORDS Coupled quantum states, Quantum mechanics
Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation prin­ciple a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.
Within the functional differential formalism of quantum systems, referred to as the quantum dynamical principle, given independent pairs of canonical conjugate variables {qi(t), pi(t),i=1,?...,?n}?{q(t), p(t)}, and a set of pairwise commuting operator functions {Gj(q(t), p(t)),j=1,?...,?k} of these variables defined, transformation functions are explicitly given, for constrained dynamical systems, expressed as functional differential operations applied to a given functional written in closed form. In the functional differential treatment external sources are, a priori, necessarily introduced in the theory. The connection of this work to the so-called Faddeev?Popov technique in path integrals is pointed out.