Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Arrival Times in Quantum Theory from an Irreversible Detector Model

J. J. Halliwell

Published Jan 1, 1998• 29 citations
OpenAlex

Abstract

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

Related papers

Arrival Times in Quantum Theory from an Irreversible Detector Model

J. J. Halliwell

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.

arXiv:quant-ph/9805057•Progress of Theoretical Physics•Oct 1, 1999•51 citations

Quantum time of arrival distribution in a simple lattice model

Shrabanti Dhar, Subinay Dasgupta, Abhishek Dhar

Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the time of arrival (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider a setup where a detector keeps making instantaneous measurements at regular finite time intervals until it detects the particle at time t , which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics.

Journal of Physics A Mathematical and Theoretical•Feb 25, 2015•68 citations

Decoherent histories approach to the arrival time problem

J. J. Halliwell, Elias Zafiris

What is the probability of a particle entering a given region of space at any time between ${t}_{1}$ and ${t}_{2}?$ Standard quantum theory assigns probabilities to alternatives at a fixed moment of time and is not immediately suited to questions of this type. We use the decoherent histories approach to quantum theory to compute the probability of a nonrelativistic particle crossing $x=0$ during an interval of time. For a system consisting of a single nonrelativistic particle, histories coarse grained according to whether or not they pass through spacetime regions are generally not decoherent, except for very special initial states, and thus probabilities cannot be assigned. Decoherence may, however, be achieved by coupling the particle to an environment consisting of a set of harmonic oscillators in a thermal bath. Probabilities for spacetime coarse grainings are thus calculated by considering restricted density operator propagators of the quantum Brownian motion model. We also show how to achieve decoherence by replicating the system $N$ times and then projecting onto the number density of particles that cross during a given time interval, and this gives an alternative expression for the crossing probability. The latter approach shows that the relative frequency for histories is approximately decoherent for sufficiently large $N,$ a result related to the Finkelstein-Graham-Hartle theorem.

arXiv:quant-ph/9706045•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 15, 1998•43 citations

Time-of-arrival probabilities for general particle detectors

Charis Anastopoulos, Ntina Savvidou

We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total system to its complementary subspace. This property makes it possible to derive a general expression for the probability for the time of transition, valid for any quantum system, with the only requirement that the time of transition is correlated with a definite macroscopic record. The framework developed here allows for the consideration of any experimental configuration for the measurement of the time of arrival, and it also applies to relativistic systems with interactions described by quantum field theory. We use the method in order to describe time-of-arrival measurements in high-energy particle reactions and for a rigorous derivation of the time-integrated probabilities in particle oscillations.

arXiv:1205.2781•Physical Review A•Jul 20, 2012•44 citations

Time-of-arrival probabilities and quantum measurements. II. Application to tunneling times

Charis Anastopoulos, Ntina Savvidou

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.

arXiv:0706.2492•Journal of Mathematical Physics•Feb 1, 2008•16 citations

Aspects of Time in Quantum Theory

James M. Yearsley

We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.

arXiv:1110.5790•Oct 1, 2011•4 citations
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.
Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the time of arrival (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider a setup where a detector keeps making instantaneous measurements at regular finite time intervals until it detects the particle at time t , which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics.
What is the probability of a particle entering a given region of space at any time between t1{t}_{1}t1​ and t2?{t}_{2}?t2​? Standard quantum theory assigns probabilities to alternatives at a fixed moment of time and is not immediately suited to questions of this type. We use the decoherent histories approach to quantum theory to compute the probability of a nonrelativistic particle crossing x=0x=0x=0 during an interval of time. For a system consisting of a single nonrelativistic particle, histories coarse grained according to whether or not they pass through spacetime regions are generally not decoherent, except for very special initial states, and thus probabilities cannot be assigned. Decoherence may, however, be achieved by coupling the particle to an environment consisting of a set of harmonic oscillators in a thermal bath. Probabilities for spacetime coarse grainings are thus calculated by considering restricted density operator propagators of the quantum Brownian motion model. We also show how to achieve decoherence by replicating the system NNN times and then projecting onto the number density of particles that cross during a given time interval, and this gives an alternative expression for the crossing probability. The latter approach shows that the relative frequency for histories is approximately decoherent for sufficiently large N,N,N, a result related to the Finkelstein-Graham-Hartle theorem.
We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total system to its complementary subspace. This property makes it possible to derive a general expression for the probability for the time of transition, valid for any quantum system, with the only requirement that the time of transition is correlated with a definite macroscopic record. The framework developed here allows for the consideration of any experimental configuration for the measurement of the time of arrival, and it also applies to relativistic systems with interactions described by quantum field theory. We use the method in order to describe time-of-arrival measurements in high-energy particle reactions and for a rigorous derivation of the time-integrated probabilities in particle oscillations.
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.
We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.