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arXiv:1808.02645 · International Journal of Modern Physics B

Exact decoherence brought by one internal degree of freedom: von Neumann equation approach and examples

John Paul A. Besagas, Jan Carlo L. Lima, Eric A. Galapon

Published Sep 30, 2019• 0 citations
arXivHTMLPDF

Abstract

In a quantum measurement setting, it is known that environment-induced decoherence theory describes the emergence of effectively classical features of the quantum system–measuring apparatus composite system when the apparatus is allowed to interact with the environment. In [E. A. Galapon, Europhys. Lett. 113, 60007 (2016)], a measurement model is found to have the feature of inducing exact decoherence at a finite time via one internal degree of freedom of the apparatus provided that the apparatus is decomposed into a pointer and an inaccessible probe, with the pointer and the probe being in momentum-limited initial states. However, an issue can be raised against the model: while the factorization method of the time-evolution operator used there is formally correct, it is not completely rigorous due to some unstated conditions on the validity of the factorization in the Hilbert space of the model. Furthermore, no examples were presented there in implementing the measurement scheme in specific quantum systems. The goal of this paper is to re-examine the model and confirm its features independently by solving the von Neumann equation for the joint state of the composite system as a function of time. This approach reproduces the joint state obtained in the original work, leading to the same conditions for exact decoherence and orthogonal pointer states when the required initial conditions on the probe and pointer are imposed. We illustrate the exact decoherence process in the measurement of observables of a spin-1/2 particle and a quantum harmonic oscillator by using the model.

DOI
10.1142/s0217979219502850

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In a quantum measurement setting, it is known that environment-induced decoherence theory describes the emergence of effectively classical features of the quantum system–measuring apparatus composite system when the apparatus is allowed to interact with the environment. In [E. A. Galapon, Europhys. Lett. 113, 60007 (2016)], a measurement model is found to have the feature of inducing exact decoherence at a finite time via one internal degree of freedom of the apparatus provided that the apparatus is decomposed into a pointer and an inaccessible probe, with the pointer and the probe being in momentum-limited initial states. However, an issue can be raised against the model: while the factorization method of the time-evolution operator used there is formally correct, it is not completely rigorous due to some unstated conditions on the validity of the factorization in the Hilbert space of the model. Furthermore, no examples were presented there in implementing the measurement scheme in specific quantum systems. The goal of this paper is to re-examine the model and confirm its features independently by solving the von Neumann equation for the joint state of the composite system as a function of time. This approach reproduces the joint state obtained in the original work, leading to the same conditions for exact decoherence and orthogonal pointer states when the required initial conditions on the probe and pointer are imposed. We illustrate the exact decoherence process in the measurement of observables of a spin-1/2 particle and a quantum harmonic oscillator by using the model.
\n\nCurrent quantum orthodoxy claims that the statistical collapse of the wave function arises from the interaction of the measuring instrument with its environment through the phenomenon known as environment-induced decoherence. Here it is shown that there exists a measurement scheme that is exactly decohering without the aid of an environment. The scheme relies on the assumption that the meter is decomposable into probe and pointer, with the probe taken to be inaccessible for observation. Under the assumption that the probe and the pointer initial states are momentum limited, it is shown that coherences die out within a finite measurement time and the pointer states are exactly orthogonal after a sufficiently longer time of measurement. Furthermore, it is shown that the measurement scheme reproduces the main result of the environment-induced decoherence theory that coherences decay asymptotically in time for a general initial state of the probe.\n
Current quantum orthodoxy claims that the statistical collapse of the wave-function arises from the interaction of the measuring instrument with its environment through the phenomenon known as environment induced decoherence. Here it is shown that there exists a measurement scheme that is exactly decohering without the aid of an environment. The scheme relies on the assumption that the meter is decomposable into probe and pointer, with the probe taken to be inaccessible for observation. Under the assumption that the probe and the pointer initial states are momentum limited, it is shown that coherences die out within a finite interval of time and the pointer states are exactly orthogonal. These lead to the fundamental realization that dispersion of correlation does not require an external infinite number of degrees of freedom. An internal one degree of freedom is already sufficient to delocalize the correlations and leave its subparts in a classical mixed state, so that decoherence may occur even for isolated measuring instruments.
Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of closed systems, we study coarse-grained decoherent histories. The system under consideration is one-dimensional and consists of a particle coupled to a von Neumann apparatus that measures its position. The particle moves in a quadratic potential; in particular we consider a driven harmonic oscillator. The real line is divided into intervals of the same length, and coarse-grained histories are defined by the arithmetic average of the initial and final position of the particle to be within one of these intervals. The position of the pointer correlates with this arithmetic average. In addition a constant term is added to this average, which is a result of the presence of a driving force on the oscillator. We investigate decoherence for such histories via the decoherence functional for the particle-apparatus. An exact expression for this functional has been derived. If the particle or the pointer of the apparatus is in an exact position state initially, then exact decoherence ensues, and the relative probability for such histories is obtained. If the initial state of the particle, which is centered at some arbitrary point in the xxx-axis, is narrow compared to the width of the intervals on the xxx-axis, then we obtain approximate decoherence. The same result follows if the initial state of the pointer is narrow. In these two situations, we evaluate expressions for the decoherence functional qualitatively and quantitatively. In the latter case we obtain the first two terms in an expansion in powers of the width of the initial state of the particle or the pointer respectively. Approximate relative probabilities can be assigned in this case, and we have obtained an expansion up to second order in powers of the width of the initial state for an expression for an upper bound of these probabilities.
Quantum decoherence is a physical process resulting from a system’s entanglement with an environment. This chapter examines the explanatory wealth of decoherence when brought to bear on foundational issues such as the question of why certain measurement bases seem to be preferred by nature – including the pointer basis of measuring apparatuses – the old familiar measurement problem, and the emergence of classicality. Decoherence satisfactorily explains the behavior of the ammonia molecule. Decoherence introduces no additional nonlinearity but “simply takes real, inevitable, interaction with external degrees of freedom to bear in the Hamiltonian”. The basis-specific nature of decoherence will allow for a satisfying explanation to one of quantum mechanics’ oldest mysteries: that of the pointer basis. The basis-specific nature of decoherence will allow for a satisfying explanation to one of quantum mechanics’ oldest mysteries: that of the pointer basis.
Abstract In quantum mechanics, pointer states are eigenstates of the observable of the measurement apparatus that represent the possible positions of the display pointer of the equipment. The origin of this concept lies in attempts to fill the blanks in Everett’s relative-state interpretation, and to make it a fully valid description of physical reality. To achieve this, it was necessary to consider not only the main system interacting with the measurement apparatus (like von Neumann and Everett did) but also the role of the environment in eliminating correlations between different possible measurements when interacting with the measurement apparatus. The interaction of the environment with the main system (and the measurement apparatus) is the core of the decoherence theory, which followed Everett’s thesis. In this article, we review the measurement process according to von Neumann, Everett’s relative state interpretation, the purpose of decoherence and some of its follow-up until Wojciech Zurek’s primordial paper that consolidated the concept of pointer states, previously presented by Heinz Dieter Zeh. Employing a simple physical model consisting of a pair of two-level systems—one representing the main system, the other the measurement apparatus—and a thermal bath—representing the environment—we show how pointer states emerge, explaining its contributions to the question of measurement in quantum mechanics, as well as its limitations. Finally, we briefly show some of its consequences. This paper is accessible to readers with elementary knowledge about quantum mechanics, on the level of graduate courses.
In a close form without referring the time-dependent Hamiltonian to the total system, a consistent approach for quantum measurement is proposed based on Zurek's triple model of quantum decoherence [W. Zurek, Phys. Rev. D 24, 1516 (1981)]. An exactly solvable model based on the intracavity system is dealt with in detail to demonstrate the central idea in our approach: by peeling off one collective variable of the measuring apparatus from its many degrees of freedom, as the pointer of the apparatus, the collective variable decouples with the internal environment formed by the effective internal variables, but still interacts with the measured system to form a triple entanglement among the measured system, the pointer, and the internal environment. As another mechanism to cause decoherence, the uncertainty of relative phase and its many-particle amplification can be summed up to an ideal entanglement or a Shmidt decomposition with respect to the preferred basis.