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 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.
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.
We study a discrete-time quantum walk in presence of a detector at
xD initially. The detector here is repeatedly removed after a span of
tR, the removal time, and reinserted at random locations. Two relocation rules are considered here: In Model~1, the detector is reinserted at any site beyond
xD, while in Model~2, reinsertion is done within a restricted window around the position of the detector at that time. Both variants behave like Semi Infinite Walk (SIW) for large
tR, where the detector behaves effectively as a fixed boundary. However, in the rapid-relocation regime, i.e., when
tR is small, the behaviours are different. Model~1 permits greater spreading due to unrestricted reinsertion, which is different from Model~2. The time evolution of occupation probability ratio of our walker to that of an infinite walker at
xD, i.e.,
f(xD,t)/f∞(xD,t), initially show the feature of a SIW upto
t=tR, then show some oscillatory behaviour and finally reach a saturation value for both the models. The ratio enhancing under certain conditions of
xD and
tR, is a purely quantum mechanical effect. The saturation ratio shows a crossover behavior below and above a removal time
tR∗. At sites
x=xD the occupation probablity ratios at a certain time reveals that for small
tR, the behaviours of the two models are drastically different from each other, as well as from Semi Infinite Walk (SIW), Quenched Quantum Walk (QQW) and Moving Detector Quantum Walk (MDQW). The correlation ratios of the two models with that of Infinite Walk (IW) show interesting time dependence for sites to the left or right of the initial detector position
xD.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier 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 in C. Anastopoulos and N. 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 for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.
Abstract How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts.