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Arrival and Passage Times From a Spin-Boson Detector Model

Jens Timo Neumann

Published Jan 1, 2007• 0 citations
Publisher

Abstract

The aim of the present thesis is twofold: On one hand it considers the treatment of time observables in quantum mechanics. In particular, it is concerned with the application of a particular spin-boson detector model to arrival and passage times. On the other hand it explores the possibility of extending the quantum jump approach to a model beyond its original, quantum optical framework. It also examines by means of numerical examples whether the quantum jump approach, which employs a continuum of bath modes, promises to provide a good approximation to situations where one actually has to deal with a number of discrete bath modes.

DOI
10.53846/goediss-2743

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Arrival and Passage Times From a Spin-Boson Detector Model

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Die vorliegende Arbeit verfolgt ein zweifaches Ziel: Einerseits befasst sie sich mit der Behandlung von Zeitobservablen in der Quantenmechanik. Insbesondere beschäftigt sie sich mit der Anwendung eines konkreten Spin-Boson Detektor-Modells auf Ankunfts- und Durchflugszeiten. Andererseits untersucht sie die Möglichkeit, die Quantensprung-Methode auf ein Modell außerhalb ihres ursprünglichen quantenoptischen Rahmens zu erweitern. Sie untersucht auch mittels numerischer Beispiele, ob die Quantensprung-Methode, die ein Kontinuum von Bad-Moden verwendet, eine gute Approximation an Situationen verspricht, in denen man eigentlich mit einer Anzahl diskreter Bad-Moden konfrontiert ist.

•Apr 14, 2011•0 citations
The aim of the present thesis is twofold: On one hand it considers the treatment of time observables in quantum mechanics. In particular, it is concerned with the application of a particular spin-boson detector model to arrival and passage times. On the other hand it explores the possibility of extending the quantum jump approach to a model beyond its original, quantum optical framework. It also examines by means of numerical examples whether the quantum jump approach, which employs a continuum of bath modes, promises to provide a good approximation to situations where one actually has to deal with a number of discrete bath modes.
The passage-time distribution for a spread-out quantum particle to traverse a specific region is calculated using a detailed quantum model for the detector involved. That model, developed and investigated in earlier works, is based on the detected particle's enhancement of the coupling between a collection of spins (in a metastable state) and their environment. We treat the continuum limit of the model, under the assumption of the Markov property, and calculate the particle state immediately after the first detection. An explicit example with 15 boson modes shows excellent agreement between the discrete model and the continuum limit. Analytical expressions for the passage-time distribution as well as numerical examples are presented. The precision of the measurement scheme is estimated and its optimization discussed. For slow particles, the precision goes like ${E}^{\ensuremath{-}3∕4}$, which improves previous ${E}^{\ensuremath{-}1}$ estimates, obtained with a quantum clock model.
We investigate a fully quantum mechanical spin model for the detection of a moving particle. This model, developed in earlier work, is based on a collection of spins at fixed locations and in a metastable state, with the particle locally enhancing the coupling of the spins to an environment of bosons. Appearance of bosons from particular spins signals the presence of the particle at the spin location, and the first boson indicates its arrival. The original model used discrete boson modes. Here we treat the continuum limit, under the assumption of the Markov property, and calculate the arrival-time distribution for a particle to reach a specific region.
We investigate a fully quantum mechanical spin model for the detection of a moving particle. This model, developed in earlier work, is based on a collection of spins at fixed locations and in a metastable state, with the particle locally enhancing the coupling of the spins to an environment of bosons. The appearance of bosons from particular spins signals the presence of the particle at the spin location, and the first boson indicates its arrival. The original model used discrete boson modes. Here we treat the continuum limit, under the assumption of the Markov property, and calculate the arrival-time distribution for a particle to reach a specific region.
In this paper we study the construction of probability densities for time of arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time of arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of the Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit—there are, however, problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.
We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit–there are however problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the timeof-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski. 1
Die vorliegende Arbeit verfolgt ein zweifaches Ziel: Einerseits befasst sie sich mit der Behandlung von Zeitobservablen in der Quantenmechanik. Insbesondere beschäftigt sie sich mit der Anwendung eines konkreten Spin-Boson Detektor-Modells auf Ankunfts- und Durchflugszeiten. Andererseits untersucht sie die Möglichkeit, die Quantensprung-Methode auf ein Modell außerhalb ihres ursprünglichen quantenoptischen Rahmens zu erweitern. Sie untersucht auch mittels numerischer Beispiele, ob die Quantensprung-Methode, die ein Kontinuum von Bad-Moden verwendet, eine gute Approximation an Situationen verspricht, in denen man eigentlich mit einer Anzahl diskreter Bad-Moden konfrontiert ist.