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arXiv:quant-ph/0610041 · Physical Review A

Passage-time distributions from a spin-boson detector model

Gerhard C. Hegerfeldt, Jens Timo Neumann, Lawrence S. Schulman

Published Jan 12, 2007• 5 citations

Abstract

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.

DOI: 10.1103/physreva.75.012108

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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.

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.

ABSTRACT: We investigate a detector scheme designed to measure the arrival of a particle at x = 0 during a finite time interval. The detector consists of a two state system which undergoes a transition from one state to the other when the particle crosses x = 0, and possesses the realistic feature that it is effectively irreversible as a result of being coupled to a large environment. The probabilities for crossing or not crossing x = 0 thereby derived coincide with earlier phenomenologically proposed expressions involving a complex potential. The probabilities are compared with similar previously proposed expressions involving sums over paths, and a connection with time operator approaches is also indicated

We investigate a detector scheme designed to measure the arrival of a particle at x = 0 during a finite time interval. The detector consists of a two-state system that undergoes a transition from one state to the other when the particle crosses x = 0, and possesses the realistic feature that it is effectively irreversible as a result of being coupled to a large environment. The probabilities for crossing or not crossing x = 0 thereby derived coincide with earlier phenomenologically proposed expressions involving a complex potential. The probabilities are compared with similar previously proposed expressions involving sums over paths, and a connection with time operator approaches is also indicated.

We study the decay of unstable states by formulating quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time-of-arrival determined by Anastopoulos and Savvidou [J. Math. Phys. 47, 122106 (2006)]. This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exponential decay only arises if three specific mathematical conditions are met. Their physical content is the following: (i) the decay time is much larger than any microscopic timescale, so that the fine details of the initial state can be ignored, (ii) there is no quantum coherence between the different “attempts” of the particle to traverse the barrier, and (iii) the transmission probability varies little within the momentum spread of the initial state. We also determine the long time limits of the decay probability and we identify regimes, in which the decays have no exponential phase.