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arXiv:2405.02018 · Physical Review A

Time-of-arrival distributions for continuous quantum systems and application to quantum backflow

Mathieu Beau, Maximilien Barbier, Rafael Martellini, Lionel Martellini

Published Nov 18, 2024• 9 citations
arXivHTMLPDF

Abstract

Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$ (position or otherwise), the distribution ${\ensuremath{\pi}}_{a}(t)$ of time measurement at a fixed state $a$ can be inferred from the distribution ${\ensuremath{\rho}}_{t}(a)$ of a state measurement at a fixed time $t$ via the transformation ${\ensuremath{\pi}}_{a}(t)\ensuremath{\propto}|\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}{\ensuremath{\int}}_{\ensuremath{-}\ensuremath{\infty}}^{a}{\ensuremath{\rho}}_{t}(u)du|$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time of arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.

DOI
10.1103/physreva.110.052217

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Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable A\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}A\ifmmode^\elseˆ\fi​ (position or otherwise), the distribution \ensuremathπa(t){\ensuremath{\pi}}_{a}(t)\ensuremathπa​(t) of time measurement at a fixed state aaa can be inferred from the distribution \ensuremathρt(a){\ensuremath{\rho}}_{t}(a)\ensuremathρt​(a) of a state measurement at a fixed time ttt via the transformation \ensuremathπa(t)\ensuremath∝∣\ensuremath∂\ensuremath∂t\ensuremath∫\ensuremath−\ensuremath∞a\ensuremathρt(u)du∣{\ensuremath{\pi}}_{a}(t)\ensuremath{\propto}|\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}{\ensuremath{\int}}_{\ensuremath{-}\ensuremath{\infty}}^{a}{\ensuremath{\rho}}_{t}(u)du|\ensuremathπa​(t)\ensuremath∝∣\ensuremath∂t\ensuremath∂​\ensuremath∫\ensuremath−\ensuremath∞a​\ensuremathρt​(u)du∣. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time of arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.
Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass mmm falling in a uniform gravitational field ggg, we use this approach to obtain an exact explicit expression for the probability density of the time of arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by $\ensuremath{\delta}=\frac{{q}^{2}}{2}$ when the factor $q\ensuremath{\equiv}\frac{\ensuremath{\hbar}}{2m\ensuremath{\sigma}\sqrt{2gx}}\ensuremath{\ll}1$ (semiclassical regime) and by $\ensuremath{\delta}=\sqrt{{\frac{2}{\ensuremath{\pi}}}^{\phantom{\ensuremath{\int}}}}q$ when $q\ensuremath{\gg}1$ (quantum regime), where $\ensuremath{\sigma}$ is the width of the initial Gaussian wave packet and xxx is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.
The time-of-arrival problem asks for the probability distribution for when a quantum particle reaches a specified location. It has been the subject of decades of debate, exemplifying the lack of a self-adjoint time observable in quantum theory. In the Page-Wootters framework, time is a relational quantity, emerging from correlations between a system and a clock induced by a global Hamiltonian constraint. We construct a time-of-arrival distribution by inverting the Page-Wootters approach, asking what time a clock reads given that the particle arrives at some fixed position. The result coincides with a common approach to the time-of-arrival problem, suggesting a potential relational interpretation of the latter. In addition to providing a relational description of the time-of-arrival problem, this gives an application of the Page-Wootters formalism to a concrete physical problem, and reveals some complications with its canonical interpretation as a theory of conditional probabilities.
Abstract The probability distribution of a time measurement Tx at position x can be inferred from the probability distribution of a position measurement Xt at time t as given by the Born rule [1, 2]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [1]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass m dropped in a uniform gravitational field g, the uncertainties about time and position measurements are related by the relation Δ T x Δ X t ≥ ℏ 2 m g This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector’s position that can be interpreted as the signature of a Zitterbewegung-like effect.
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.
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 \emph{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 the setup where a detector keeps making instantaneous measurements at regular finite time intervals {\emph{till}} it detects the particle at some time ttt, 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.