J. G. Muga, R. Sala, J. P. Palao
The concept and the formalization of the arrival time in quantum mechanics are discussed. Different approaches based on trajectories, quantization rules, time operators, phase space techniques, renewal equations or operational procedures are reviewed or proposed. Open questions and loose ends are pointed out.
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Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle-antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory.
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.
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.
Arbitrarily precise arrival time measurements in quantum mechanics
本文通过局部检测过程和适当选择的边界条件构建了任意精确的到达时间测量方案，并证明在极限下吸收边界条件可描述粒子与探测器的相互作用。
The time of arrival at an arbitrary position in configuration space can be given as a function of the phase space variables for the Liouville integrable systems of classical mechanics, but only for them. We review the Jacobi-Lie transformation that explicitly implements this function of positions and momenta. We then discuss the recently developed quantum formalism for the time of arrival. We first analyze the case of free particles in one and three space dimensions. Then, we apply the quantum version of the Jacobi-Lie transformation to work out the time of arrival operator in the presence of …
Arrival time in quantum mechanics (demonstrated in geometrical order)
该研究通过分析经典可观测量的不同量子化程序，提出量子力学中到达时间的几何构造方法，给出唯一量子化版本可观测量的存在判据并应用于经典到达时间可观测量。
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…
We propose a covariant algorithm for relativistic ideal measurements and for relativistic continuous measurements, its non-relativistic limit results the algorithm of the Event-Enhanced Quantum Theory. Therefore an additional intrinsic parameter, the proper time, is used. As an application we compute the time of arrival of a particle at a detector and find good agreement between the expected values of the time of arrival for weak detectors and the results of the relativistic point-mechanic over a wide range. For very high momentums there is a small probability for a negative time of arrival, s…
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.
Assorted questions: Time as a parameter in Quantum Mechanics. No-Go theorems for a time operator. Localization, time and causality. Causality violation. Localization again. Lesson 1: Evading the troubles: Im E finite. Lights and shadows of a time operator:"Table-top Spacetime" quantum mechanics. Biphotons at Berkeley. Time operator build-up. Good news - bad news. Lesson 2: We need the resonances to tell one.
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。
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…
We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schrödinger curren…
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.
It is well known that orthodox quantum mechanics does not make unambiguous predictions for the statistics in arrival time (or time-of-flight) experiments. Bohmian mechanics (or de Broglie-Bohm theory) offers a distinct conceptual advantage in this regard, owing to the well defined concepts of point particles and trajectories embedded in this theory. We revisit a recently proposed experiment [S. Das and D. Dürr, Sci. Rep. (2019)], the numerical analysis of which revealed a striking spin dependence in the (Bohmian) time-of-arrival distributions of a spin-1/2 particle. We present here a mathemati…
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 introduced with coauthors some years ago a solution to the problem of time in quantum gravity which consists in formulating the quantum theory in terms of real clocks. It combines Page and Wootters' relational proposal with Rovelli's evolving constants of the motion. Time is associated with an operator and not a classical parameter. We show here that this construction provides a natural solution to the time of arrival problem in quantum mechanics and leads to a well defined time-energy uncertainty relation for the clocks.
We extend the concept of confined quantum time of arrival operators, first developed for the free particle [E.A. Galapon, R. Caballar, R. Bahague {\it Phys. Rev. Let.} {\bf 93} 180406 (2004)], to arbitrary potentials.
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 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.
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.
Arrival time and Bohmian Mechanics: It is the theory which decides what we can measure
本研究围绕玻姆力学框架下量子粒子到达时间分布测量的相关方案与学术争议展开，分析争论双方的正误，明确该方案的可测量性及与无信号定理的关系。
We extend previous work on the arrival time problem in quantum mechanics, in the framework of decoherent histories, to the case of a particle coupled to an environment. The usual arrival time probabilities are related to the probability current, so we explore the properties of the current for general open systems that can be written in terms of a master equation of Lindblad form. We specialise to the case of quantum Brownian motion, and show that after a time of order the localisation time the current becomes positive. We show that the arrival time probabilities can then be written in terms of…
We compute in a relativistic way the time-of-arrival and the traversal time through a region of a free particle with spin 1/2. We do this by applying the relativistic extension of the Event-Enhanced Quantum Theory which we have presented in a previous paper. We find a very good coincidence of the results of our formalism and the results obtained by using classical relativistic mechanics.
An operational time of arrival is introduced using a realistic position and momentum measurement scheme. The phase space measurement involves the dynamics of a quantum particle probed by a measuring device. For such a measurement an operational positive operator valued measure in phase space is introduced and investigated. In such an operational formalism a quantum mechanical time operator is constructed and analyzed. A phase space time and energy uncertainty relation is derived.
Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.
New prescription to treat position and time equally in quantum mechanics is presented. Using this prescription, we could successfully derive some interesting formulae such as time-of-arrival for a free particle and quantum tunneling formula. The physical interpretation will be discussed.
We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schrödinger's equation does not naturally define a probability density at the continuum limit, but also becau…
Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by J. Kijowski and the EEQT method. It is suggested that for free particles the two methods may lead to the same result. On the other hand the EEQT method can be easily geometrized within the framework of Galilei-Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.
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.
Using the concept of crossing state and the formalism of second quantization, we propose a prescription for computing the density of arrivals of particles for multiparticle states, both in the free and the interacting case. The densities thus computed are positive, covariant in time for time independent hamiltonians, normalized to the total number of arrivals, and related to the flux. We investigate the behaviour of this prescriptions for bosons and fermions, finding boson enhancement and fermion depletion of arrivals.
We raise the possibility of developing a theory of constructing quantum dynamical observables independent from quantization and deriving classical dynamical observables from pure quantum mechanical consideration. We do so by giving a detailed quantum mechanical derivation of the classical time of arrival at arbitrary arrival points for a particle in one dimension.
It has always been believed that no self-adjoint and canonical time of arrival operator can be constructed within the confines of standard quantum mechanics. In this Letter we demonstrate the otherwise. We do so by pointing out that there is no a priori reason in demanding that canonical pairs form a system of imprimitivities. We then proceed to show that a class of self-adjoint and canonical time of arrival (TOA) operators can be constructed for a spatially confined free particle. And then discuss the relatiobship between the non-self-adjointess of the TOA operator for the unconfined particle…
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…
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。
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…
A nonreflecting wavepacket is constructed by the superposition of reflectionless eigenstates of sech2 potential. Free propagation and propagation in the presence of the above potential of such a wavepacket is considered using the concept of arrival time. Comparison is made with the free evolving Gaussian wavepacket. Mean arrival time at a detector behind the well is given as a function of mass for separate cases. A selection of Bohmian trajectories in the interacting case are computed and compared to the trajectories of a free particle guided by a Gaussian packet.
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-…
A solution of the quantum time of arrival problem via mathematical probability theory
本文通过数学概率理论解决了量子到达时间问题，构建了理想探测器模型，并探讨了其在几何量子理论中的应用及未来研究方向。
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 review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expres…
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.
We examine the measurability of the temporal ordering of two events, as well as event coincidences. In classical mechanics, a measurement of the order-of-arrival of two particles is shown to be equivalent to a measurement involving only one particle (in higher dimensions). In quantum mechanics, we find that diffraction effects introduce a minimum inaccuracy to which the temporal order-of-arrival can be determined unambiguously. The minimum inaccuracy of the measurement is given by dt=1/E where E is the total kinetic energy of the two particles. Similar restrictions apply to the case of coincid…
In the arrival time problem in quantum mechanics, a standard formula that frequently emerges as the probability for crossing the origin during a given time interval is the current integrated over that time interval. This is semiclassically correct but can be negative due to backflow. Here, we show that this formula naturally arises in a decoherent histories analysis of the arrival time problem. For a variety of initial states, we show that histories crossing during different time intervals are approximately decoherent. Probabilities may therefore be assigned and coincide with the standard form…
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…
The model of weak measurements is applied to various problems, related to the time problem in quantum mechanics. The review and generalization of the theoretical analysis of the time problem in quantum mechanics based on the concept of weak measurements are presented. A question of the time interval the system spends in the specified state, when the final state of the system is given, is raised. Using the concept of weak measurements the expression for such time is obtained. The results are applied to the tunneling problem. A procedure for the calculation of the asymptotic tunneling and reflec…
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。
We present a decomposition of the general quantum mechanical evolution operator, that corresponds to the path decomposition expansion, and interpret its constituents in terms of the quantum Zeno effect (QZE). This decomposition is applied to a finite dimensional example and to the case of a free particle in the real line, where the possibility of boundary conditions more general than those hitherto considered in the literature is shown. We reinterpret the assignment of consistent probabilities to different regions of spacetime in terms of the QZE. The comparison of the approach of consistent h…
Possible theoretical frameworks for measurement of (arrival) time in the nonrelativistic quantum mechanics are reviewed. It is argued that the ambiguity between indirect measurements by a suitably introduced time operator and direct measurements by a physical clock particle has a counterpart in the corresponding classical framework of measurement of the Newtonian time based on the Hamiltonian mechanics.
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。
It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin-1/2 particles. In the nonrelativistic domain this implies a guidance law for electrons which differs by an additional spin-dependent term from the one originally proposed by de Broglie and Bohm. Although the additional term in the guidance equation may not be detectable in the quantum measurements derived solely from the probability density $ρ$, it plays a role in the case of arrival-time measurements. In this paper we compute the arrival…
Space-time-symmetric extension of quantum mechanics: Interpretation and arrival-time predictions
本文解释量子力学的时空对称扩展，阐明空间条件薛定谔方程、动量本征态及到达时间概率幅，并指出位置态与时间态对同一粒子提供互补信息。
A Bohmian analysis of the so-called Schrödinger-Langevin or Kostin nonlinear differential equation is provided to study how thermal fluctuations of the environment affects the dynamics of the wave packet from a quantum hydrodynamical point of view. In this way, after obtaining the Schrödinger-Langevin-Bohm equation from the Kostin equation its application to simple but physically insightful systems such as the Brownian-Bohmian motion, motion in a gravity field and transmission through a parabolic repeller is studied. % If a time-dependent Gaussian ansatz for the probability density is assumed,…
The weak equivalence principle (WEP) in the quantum regime has been the subject of many studies with a broad range of approach to the problem. Here we tackle the problem anew through the time of arrival (TOA) operator approach by constructing the time of arrival operator for a non-relativistic and structureless particle that is projected upward in a uniform gravitational field with an intended arrival point below the classical turning point. The TOA-operator is constructed under the constraint that the inertial and gravitational masses are equivalent, and that Galilean invariance is preserved.…
The well-known Hong-Ou-Mandel effect is revisited. Two physical reasons are discussed for the effect to be less pronounced or even to disappear: differing polarizations of photons coming to the beamsplitter and delay time of photons in one of two channels. For the latter we use the concepts of biphoton frequency and temporal wave functions depending, correspondingly, on two frequency continuous variables of photons and on two time variables $t_1$ and $t_2$ interpreted as the arrival times of photons to the beamsplitter. Explicit expressions are found for the probability densities and total pro…
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。
Thermal wave packets are used to analyze transmission probabilities and characteristic times through a parabolic repeller within the dissipative Bohmian mechanics. Thermal arrival, dwelling, transmission and reflection times are defined and calculated by using a Maxwell-Boltzmann distribution for the initial velocities of the incident particles. The dissipation is considered within the Caldirola-Kanai and Kostin approaches where a linear and nonlinear framework is used, respectively. The initial parameters are chosen to have only a dissipative tunnelling dynamics at zero temperature; at any no…
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 …
Testing trajectory-based determinism via time probability distributions
本文通过构造到达时间概率分布并推导量子力学无法达到的条件概率分布，提出一种无需直接测量到达时间即可实验检验基于轨迹的决定论有效性的方法。
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