J. Finkelstein
We consider the definition that might be given to the time at which a particle arrives at a given place, both in standard quantum theory and also in Bohmian mechanics. We discuss an ambiguity that arises in the standard theory in three, but not in one, spatial dimension.
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For the special case of freely evolving Dirac electrons in $1 + 1$ dimensions, Feynman checkerboard paths have previously been used to derive Wigner's arrival-time distribution which includes all arrivals. Here, an attempt is made to use these paths to determine the corresponding distribution of first-arrival times. Simple analytic expressions are obtained for the relevant components of the first-arrival propagator. These are used to investigate the relative importance of the first-arrival contribution to the Wigner arrival-time distribution and of the contribution arising from interference be…
We show that the Kijowski distribution for time of arrivals in the entire real line is the limiting distribution of the time of arrival distribution in a confining box as its length increases to infinity. The dynamics of the confined time of arrival eigenfunctions is also numerically investigated and demonstrated that the eigenfunctions evolve to have point supports at the arrival point at their respective eigenvalues in the limit of arbitrarilly large confining lengths, giving insight into the ideal physical content of the Kijowsky distribution.
It is shown that a class of exponentially decaying time-of-arrival probability distributions suggested by Włodarz, Marchewka and Schuss, and Jurman and Nikolić, as well as a semiclassical distribution implicit in time-of-flight momentum measurements, do not show the expected behavior for a Gaussian wave train. This casts doubts on the physical adequacy of these arrival-time proposals. In contrast, the quantum flux distribution (a special case of the Bohmian arrival-time distribution) displays the expected behavior.
An operational arrival-time distribution is defined as the distribution of detection times of the first photons emitted by two level atoms in resonance with a perpendicular laser beam in a time of flight experiment. For ultracold Cesium atoms the simulations are in excellent agreement with the theoretical ideal time-of-arrival distribution of Kijowski.
The arrival time statistics of spin-1/2 particles governed by Pauli's equation, and defined by their Bohmian trajectories, show unexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from either the usual (convective) quantum flux or from semiclassical considerations suggest testing the notable deviations in an arrival time experiment, thereby probing the predictive power of Bohmian trajectories. The suggested experiment, including the preparation of the wave functions, could be done with present-day experimental technology.
A realization of the concept of "crossing state" invoked, but not implemented, by Wigner, allows to advance in two important aspects of the time of arrival in quantum mechanics: (i) For free motion, we find that the limitations described by Aharonov et al. in Phys. Rev. A 57, 4130 (1998) for the time-of-arrival uncertainty at low energies for certain mesurement models are in fact already present in the intrinsic time-of-arrival distribution of Kijowski; (ii) We have also found a covariant generalization of this distribution for arbitrary potentials and positions.
We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to s…
An experimentally realizable scheme is formulated which can test any postulated quantum mechanical approach for calculating the arrival time distribution. This is specifically illustrated by using the modulus of the probability current density for calculating the arrival time distribution of spin-1/2 neutral particles at the exit point of a spin-rotator(SR) which contains a constant magnetic field. Such a calculated time distribution is then used for evaluating the distribution of spin orientations along different directions for these particles emerging from the SR. Based on this, the result o…
Several proposals for a time-of-arrival distribution of ensembles of independent quantum particles subject to an external interaction potential are compared making use of the ``crossing state'' concept. It is shown that only one of them has the properties expected for a classical distribution in the classical limit. The comparison is illustrated numerically with a collision of a Gaussian wave packet with an opaque square barrier.
We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov-Bohm and Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski's result is pointed out, along with an inescapable consequence of the "negative arrival times" inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time setup featuring a charged particle moving in a constant m…
The mean arrival time of free particles is computed using the quantum probability current. This is uniquely determined in the non-relativistic limit of Dirac equation, although the Schroedinger probability current has an inherent non-uniqueness. Since the Dirac probability current involves a spin-dependent term, an arrival time distribution based on the probability current shows an observable spin-dependent effect, even for free particles. This arises essentially from relativistic quantum dynamics, but persists even in the non-relativistic regime.
The classical limit problem of quantum mechanics is revisited on the basis of a scheme that enables a quantitative study of the way the quantum-classical agreement emerges while going through the intermediate mass range between the microscopic and the macroscopic domains. As a specific application of such a scheme, we investigate the classical limit of a quantum time distribution - an area of study that has remained largely unexplored. For this purpose, we focus on the arrival time distribution in order to examine the way the observable results pertaining to the quantum arrival time distributi…
We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. {\bf 122}, 1649 (1961). We use Naimark's dilation theorem to reproduce the generalized decomposition of unity (or POVM) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the…
A recent approach to arrival times used the fluorescence of an atom entering a laser illuminated region and the resulting arrival-time distribution was close to the axiomatic distribution of Kijowski, but not exactly equal, neither in limiting cases nor after compensation of reflection losses by normalization on the level of expectation values. In this paper we employ a normalization on the level of operators, recently proposed in a slightly different context. We show that in this case the axiomatic arrival time distribution of Kijowski is recovered as a limiting case. In addition, it is shown…
We introduce an arrival time operator which is self-adjoint and, unlike previously proposed arrival time operators, has a close link to simple measurement models. Its spectrum leads to an arrival time distribution which is a variant of the Kijowski distribution (a re-ordering of the current) in the large momentum regime but is proportional to the kinetic energy density in the small momentum regime, in agreement with measurement models. A brief derivation of the latter distribution is given. We make some simple observations about the physical reasons for self-adjointness, or its absence, in arr…
The current density for a freely evolving state without negative momentum components can temporarily be negative. The operational arrival time distribution, defined by the absorption rate of an ideal detector, is calculated for a model detector and compared with recently proposed distributions. Counterintuitive features of the backflow regime are discussed.
Arrival-time distributions as a probe of the preferred foliation in relativistic Bohmian mechanics
本文提出通过到达时间分布实验检测相对论布里渊力学中的优先分层，揭示其时间顺序对实验结果的影响，可能实现超光速信号。
In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator $\hat {\cal T}(X)$ whose eigenstates can be used to define consistently a probability distribution of the time of arrival at a given spatial point. In the present work we show that the probability distribution previously proposed can be well understood on classical grounds in the sense that it is given by the expectation value of a certain positive definite operator $\hat J^{(+)}(X)$ which is nothing but a straightforward quantum version of the modulus of the classical current. F…
A relativistic version of the Aharonov-Bohm time of arrival operator for spin-0 particles was constructed by Razavi in [Il Nuovo Cimento B \textbf{63}, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the time of arrival problem that goes beyond Razavi's original results. This allows us to use time of arrival eigenfunctions that exhibit unitary arrival to construct time of arrival distributions. The expectation value is also calculated and shown that particl…
The position-momentum quasi-distribution obtained from an Arthurs and Kelly joint measurement model is used to obtain indirectly an ``operational'' time-of-arrival (TOA) distribution following a quantization procedure proposed by Kochański and Wódkiewicz [Phys. Rev. A 60, 2689 (1999)]. This TOA distribution is not time covariant. The procedure is generalized by using other phase-space quasi-distributions, and sufficient conditions are provided for time covariance that limit the possible phase-space quasi-distributions essentially to the Wigner function, which, however, provides a non-positive …
This paper compares the proposal made in previous papers for a quantum probability distribution of the time of arrival at a certain point with the corresponding proposal based on the probability current density. Quantitative differences between the two formulations are examined analytically and numerically with the aim of establishing conditions under which the proposals might be tested by experiment. It is found that quantum regime conditions produce the biggest differences between the formulations which are otherwise near indistinguishable. These results indicate that in order to discriminat…
A number of approaches to the arrival time problem employ a complex potential of a simple step function type and the arrival time distribution may then be calculated using the stationary scattering wave functions. Here, it is shown that in the Zeno limit (in which the potential becomes very large), the arrival time distribution may be obtained in a clear and simple way using a path integral representation of the propagator together with the path decomposition expansion (in which the propagator is factored across a surface of constant time). This method also shows that the same result is obtain…
We model ideal arrival-time measurements for free quantum particles and for particles subject to an external interaction by means of a narrow and weak absorbing potential. This approach is related to the operational approach of measuring the first photon emitted from a two-level atom illuminated by a laser. By operator-normalizing the resulting time-of-arrival distribution, a distribution is obtained which for freely moving particles not only recovers the axiomatically derived distribution of Kijowski for states with purely positive momenta but is also applicable to general momentum components…
We introduce a formalism for the calculation of the time of arrival t at a space point for particles traveling through interacting media. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to construct t in the context of the Positive Operator Valued Measures. We then compute the probability distribution in the times of arrival at a point for particles that have undergone reflection, transmission or tunneling off finite potential barriers. For narrow Gaussian initial wave packets we obtain multimodal time distributions of the …
A self-adjoint operator with dimensions of time is explicitly constructed, and it is shown that its complete and orthonormal set of eigenstates can be used to define consistently a probability distribution of the time of arrival at a spatial point.
Some unexplored decoherence aspects within the Caldeira-Leggett master equation are analyzed and discussed. The decoherence process is controlled by the two environment parameters, the relaxation rate or friction and the temperature, leading to a gradual transition from the quantum to classical regime. Arrival time distributions, nonminimum-uncertainty-product or stretching Gaussian wave packets, identical particles and diffraction in time display interesting features during the decoherence process undergone by the time dependent interference patterns. We show that the presence of a constant f…
All covariant time operators with normalized probability distribution are derived. Symmetry criteria are invoked to arrive at a unique expression for a given Hamiltonian. As an application, a well known result for the arrival time distribution of a free particle is generalized and extended. Interestingly, the resulting arrival time distribution operator is connected to a particular, positive, quantization of the classical current. For particles in a potential we also introduce and study the notion of conditional arrival-time distribution.
On the Spin Dependence of Detection Times and the Nonmeasurability of Arrival Times
本研究基于量子力学中POVM支配实验结果统计的原理，论证Das与Dürr提出的带自旋相关性的到达时间分布因不符合POVM要求，对应的到达时间测量实验不可能存在。
The arrival time probability distribution is defined by analogy with the classical mechanics. The difficulty of requirement to have the values of non-commuting operators is circumvented using the concept of weak measurements. The proposed procedure is suitable to the free particles and to the particles subjected to an external potential, as well. It is shown that such an approach imposes an inherent limitation to the accuracy of the arrival time determination.
For a quantum-mechanically spread-out particle we investigate a method for determining its arrival time at a specific location. The procedure is based on the emission of a first photon from a two-level system moving into a laser-illuminated region. The resulting temporal distribution is explicitly calculated for the one-dimensional case and compared with axiomatically proposed expressions. As a main result we show that by means of a deconvolution one obtains the well known quantum mechanical probability flux of the particle at the location as a limiting distribution.
The analysis of the model quantum clocks proposed by Aharonov et al. [Phys. Rev. A 57 (1998) 4130 - quant-ph/9709031] requires considering evanescent components, previously ignored. We also clarify the meaning of the operational time of arrival distribution which had been investigated.
We further develop the general theory of quantum time distributions introduced in arXiv:2010.07575 and apply it to find the distribution of arrival times at the detector. Even though the Hamiltonian in the absence of detector is hermitian, the time evolution of the system before detection involves dealing with a non-hermitian operator obtained from the projection of the hermitian Hamiltonian onto the region in front of the detector. Such a formalism eventually gives rise to a simple and physically sensible analytical expression for the arrival time distribution, for arbitrary wave packet movin…
The Unidirectional Current as First Arrival-Time POVM: An MS-Kijowski Identity, Physical Interpretation, and Mathematical Applications
该研究将MS单向首到达电流与Kijowski到达时间分布关联，构建到达时间POVM，揭示两种量子到达时间方法的等价性，结合探测器动力学与正算子测量结构。
Comment on "the Spin Dependence of Detection Times and the Nonmeasurability of Arrival Times"
本文是对Goldstein等人关于自旋依赖到达时间分布不可测性评论的回应，针对Das和Dürr此前报道的结果进行辩护与澄清。
Arrival Times Versus Detection Times
针对量子探测时间概率分布的争议问题，本文指出玻姆力学中无探测器的到达时间与有探测器的到达时间、探测时间未必一致，并纠正了Das和Dürr相关研究中基于三者重合假设的分布结论偏差。
Detlef Dürr, arrival-time distributions, and spin in Bohmian mechanics: Personal recollections and state-of-the-art
本文回顾作者与 Detlef Dürr 的合作，围绕量子力学到达时间问题，重点梳理 Bohmian 粒子及自旋极化电子在圆柱形波导中的奇异到达时间分布及其思想背景。
We address a number of aspects of the arrival time problem defined using a complex potential of step function form. We concentrate on the limit of a weak potential, in which the resulting arrival time distribution function is closely related to the quantum-mechanical current. We first consider the analagous classical arrival time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. In the quantum case, we review the path decomposition expansion (PDX), in which the propagator is factored across a surface of constant time, so is very useful …
Measuring the arrival time of an electron wave packet using a dynamical potential barrier
本文研究时变势垒测量热电子波包到达时间时因有限上升速度导致的误差，区分半经典与准静态区域并给出精确检测的时间尺度条件。
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 so…
The present article focuses on studying the extent to which the nonuniqueness that is inherent in the standard quantum mechanical calculation of arrival/transit time distribution can be exploited to enable an empirical scrutiny of any causal trajectory model such as the Bohmian scheme. For this purpose, we consider the example of spin-1/2 neutral particles corresponding to a wave packet which passes through a spin rotator(SR) that contains constant magnetic field confined within a region - in such a case, the transit time distribution can be measured in terms of the spin distribution of partic…
Repeated measurements of a quantum particle to check its presence in a region of space was proposed long ago [G. R. Allcock, Ann. Phys. {\bf 53}, 286 (1969)] as a natural way to determine the distribution of times of arrival at the orthogonal subspace, but the method was discarded because of the quantum Zeno effect: in the limit of very frequent measurements the wave function is reflected and remains in the original subspace. We show that by normalizing the small bits of arriving (removed) norm, an ideal time distribution emerges in correspondence with a classical local-kinetic-energy distribu…
What Bohmian mechanic says about arrival times of 1D vacuum squeezed states
本文基于玻姆力学形式体系，利用薛定谔表示下的一维真空压缩态波函数与辛群幺正表示，计算了量子粒子的到达时间概率分布，并得出区别于标准量子力学的闭合表达式。
The arrival position problem in quantum mechanics
研究量子理论中关于预测探测事件发生位置分布的到达位置问题，通过比较从突出的屏幕问题解决方案得出的定量到达位置预测进行研究，指出即使在当前技术可实现的简单实验及远场极限下，不同模型预测也有差异。
The time of arrival problem in the Page-Wootters formalism
本文探讨了量子粒子到达指定位置的概率分布问题，通过反向应用页-沃特茨框架，提出了时间观测的关联解释，并揭示了该范式在条件概率解释中的复杂性。
Arrival Time\textemdash Classical Parameter or Quantum Operator?
本文探讨了量子力学中到达时间分布的解释问题，通过扩展时间参数和时间算符方法，提出双粒子实验并评估联合分布，揭示两种方法在特定条件下的不等价预测，为时间域纠缠量子技术的发展提供新见解。
Inequivalence of stochastic and Bohmian arrival times in time-of-flight experiments
本文研究超冷原子飞行时间实验中随机诠释与玻姆诠释所预言的到达时间分布，发现二者一般不同，表明基于轨迹的量子力学诠释可被实验区分。
Non-local interference in arrival time
本文提出利用纠缠原子对的改进版双双缝实验，基于Bohmian方法在数值上证明了到达时间分布中单粒子与双粒子干涉可见度的互补关系，为严格测试Bohmian到达时间分布提供了途径。
It is argued that the time-of-arrival cannot be precisely defined and measured in quantum mechanics. By constructing explicit toy models of a measurement, we show that for a free particle it cannot be measured more accurately then $Δt_A \sim 1/E_k$, where $E_k$ is the initial kinetic energy of the particle. With a better accuracy, particles reflect off the measuring device, and the resulting probability distribution becomes distorted. It is shown that a time-of-arrival operator cannot exist, and that approximate time-of-arrival operators do not correspond to the measurements considered here.
We study the effect of quantum statistics on the arrival time distribution of quantum particles computed through the probability current density. It is shown that symmetrization or asymmetrization of the wave function affects the arrival time distribution for even freely propagating particles. In order to investigate the effect of statistics on the weak equivalence principle in quantum mechanics (WEQ), we then compute the mean arrival time for wave packets in free fall. The violation of WEQ through the effect of statistics on the mass-dependence of the mean arrival time is clearly exhibited. W…
We develop a new conception for the quantum mechanical arrival time distribution from the perspective of Bohmian mechanics. A detection probability for detectors sensitive to quite arbitrary spacetime domains is formulated. Basic positivity and monotonicity properties are established. We show that our detection probability improves and generalises earlier proposals by Leavens and McKinnon. The difference between the two notions is illustrated through application to a free wave packet.
Time-of-arrival distributions for continuous quantum systems and application to quantum backflow
本文利用统计学结果证明，任意连续量子系统的到达时间分布可由固定时刻状态分布经积分求导变换得到，表明到达时间问题已隐含于玻恩规则中，无需引入时间算符或特定本体论，并展示了其在自由粒子、自由下落粒子及量子回流实验设计中的应用。
Fisher information from quantum many-particle arrival time measurements
本文提出玻色多粒子系统到达时间测量模型，推导Fisher信息，证明单粒子动量可从到达时间数据中识别，即使在稀疏束极限下也有效。
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