diff --git "a/-tE0T4oBgHgl3EQfxAG2/content/tmp_files/load_file.txt" "b/-tE0T4oBgHgl3EQfxAG2/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-tE0T4oBgHgl3EQfxAG2/content/tmp_files/load_file.txt" @@ -0,0 +1,1818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf,len=1817 +page_content='Can the double-slit experiment distinguish between quantum interpretations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Ali Ayatollah Rafsanjani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' ∗ MohammadJavad Kazemi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' † Alireza Bahrampour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 3 and Mehdi Golshani2 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Sharif University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Iran 2School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Institute for Research in Fundamental Sciences (IPM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Iran 3Research center for quantum engineering and photonics technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Sharif University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Iran Despite the astonishing successes of quantum mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' due to some fundamental problems such as the measurement problem and quantum arrival time problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the predictions of the theory are in some cases not quite clear and unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Especially, there are various predictions for the joint spatiotemporal distribution of particle detection events on a screen, which are derived from different formulations and interpretations of the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Although the differences are typically small, our studies show that these predictions can be experimentally distinguished by an unconventional double-slit configuration, which is realizable using present-day single-atom interferometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' INTRODUCTION In textbook quantum theory, time is a parameter in the Schr¨odinger equation, not a self-adjoint operator, hence there is no unique and unambiguous way to compute the temporal probability distribution of events from the first principles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the Born rule) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, since clocks exist and time measurements are routinely per- formed in quantum experiments [2, 3], a complete quan- tum theory must be able to predict the temporal statis- tics of detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' For example, in the famous dou- ble slit experiment, each particle is detected at a ran- dom time as same as at a random position on the de- tection screen [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore, one can ask: What is the position-time joint probability density P(x, t) on the screen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Although this question is very old [9–12], it is still open [13–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, the ambiguity in the arrival time distribution even prevents a clear prediction of cu- mulative arrival position distribution, � P(x, t)dt, which is typically measured in a non-time-resolved double-slit experiment [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, usual experiments are performed in the far-field (or scattering) regime, where a semiclassical analysis is often sufficient [13, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this analysis, it is assumed that particles move along classical trajectories, and the arrival time distribution is computed using the quantum momentum distribution [8, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, because of the quantum backflow effect [22], even in free space, the quantum mechanical time evolution of position probability density is not consistent with the underlying uniform motion assumption, especially in near-field inter- ference phenomena [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, due to recent progress in the ultra-fast detectors technology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' see [24–27]), it will be soon possible to investigate the near-field regime, where the semiclassical approximation breaks down and a deeper analysis would be demanded [13, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To remedy this problem, based on various interpre- tations and formulations of quantum theory, several at- tempts have been made to introduce a suitable arrival ∗ aliayat@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='edu † kazemi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='m@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='com time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' On the one hand, according to the (generalized) standard canonical interpretation, the ar- rival distribution is considered as a generalized observ- able, which is described by a positive-operator-valued measure (POVM), satisfying some required symmetries [10, 11, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' On the other hand, in the realistic- trajectory-based formulations of quantum theory, such as the Bohmian mechanics [32], Nelson stochastic me- chanics [33], and many interacting worlds interpretation [34], the arrival time distribution could be obtained from particles trajectories [7, 18, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, in other approaches, the arrival time distribution is computed via phenomenological modeling of the detection process, such as the (generalized) path integral formalism in the pres- ence of an absorbing boundary [12, 37–39], Schr¨odinger equation with complex potential or absorbing boundary [40–44], and so on [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In principle, the results of these approaches are dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, in most of the experimental situations, the differences are typically slight, and so far as we know, in the situation where differences are significant, none of the proposals have been backed up by experiments in a strict manner [8, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' An experiment that can probe these differences would undoubtedly enrich our understanding of the foundations of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The purpose of the present paper is to make it evident, via numerical simulations, that the famous two-slit experiment could be utilized to investigate these differences if we simply use a horizontal screen instead of a vertical one: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using current laser cooling and magneto-optical trapping technologies, this type of experiment can be realized by Bose-Einstein condensates, as a controllable source of co- herent matter waves [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, our numerical study shows that the required space-time resolution in particle detection is achievable using fast single-atom de- tectors, such as the recent delay-line detectors described in [51, 52] or the detector used in [6, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The structure of this paper is as follows: In Section II, we study the main proposed intrinsic arrival distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Then, in section III we compare them in the double-slit setup with vertical and horizontal screens and in different detection schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Section IV, we study the screen back-effect, and we summarize in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='02641v1 [quant-ph] 6 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' “INTRINSIC” ARRIVAL DISTRIBUTIONS In this section, we first review the semi-classical ap- proximation and then scrutinize two main proposed in- trinsic arrival time distributions [16, 36] and their asso- ciated screen observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In these approaches, the effect of the detector’s presence on the wave function evolution, before particle detection, is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We discuss this effect in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Semiclassical approximation As mentioned, in the experiments in which the detec- tors are placed far away from the support of the initial wave function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' far-field regime), the semiclassical arrival time distribution is routinely used to the descrip- tion of the particle time-of-flight [21, 54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this ap- proximation, it is assumed that particles move classically between the preparation and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this ap- proach, the arrival time randomness is understood as a result of the uncertainty of momentum, and so the arrival time distribution is obtained from momentum distribu- tion [13, 17, 36, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the one-dimensional case, the classical arrival time is given by t = m(L − x0)/p0, (1) which is applicable for a freely moving particle of mass m that at the initial t = 0 had position x0 and momen- tum p0 arriving at a distant point L on a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Hence, for a particle with the momentum wave fuction ˜ψ0(p), assuming ∆x0 ≪|L − ⟨x⟩0|, the semiclassical arrival time distribution reads [58] ΠSC(t|x=L) = mL t2 | ˜ψ0(mL/t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (2) This analysis could be generalized in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Then, the distribution of arrival time at a screen surface S is given by [36] ΠSC(t|x∈S) = m3 t4 � S | ˜ψ0(mx/t)|2 x · dS, (3) where the dS is the surface element directed outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The other main distribution that should be demanded is the joint position-time probability distribution on the screen, which is also called ”screen observable” [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Us- ing the conditional probability definition, the joint prob- ability of finding the particle in dS and in a time in- terval [t, t+dt] could be written as P(x, t|x ∈ S)dSdt = [Π(t|x∈S)dt] × [P(x|x∈S, t)dS] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, one can use the fact that ψt(x) is the state of the system, con- ditioned on the time being t in the Schr¨odinger picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This implies that |ψt(x)|2 refers to the position probabil- ity density conditioned at a specific time t [14, 15, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore, in the semiclassical approximation, the joint spatiotemporal probability density reads as PSC(x, t|x∈S) = NSCΠSC(t|x∈S) |ψt(x)|2 (4) in which NSC ≡1/ � S dS |ψt(x)|2 is the normalization con- stant, and dS =n·dS, where n is the outward unit normal vector at x∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' “Standard” approach The first attempts to investigate the arrival time prob- lem, based on the standard rules of quantum theory, were made at the beginning of the 1960s by Aharonov and Bohm [60], and also Paul [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This approach starts with a symmetric quantization of classical arrival time expres- sion (1), as follows [62]: ˆtAB = mL ˆp −1 − m 2 (ˆp −1 ˆx + ˆx ˆp −1), (5) where ˆx and ˆp=−i ∂/∂x are the usual position and mo- mentum operators, respectively, and ˆtAB is called the Aharonov-Bohm time operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This operator satisfies the canonical commutation relation with the free Hamil- tonian operator, [ˆtAB, ˆp2/2m] = iℏ, which has been used to establish the energy-time uncertainty relation [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, although ˆtAB is Hermitian (or symmetric in mathematics literature), it is not a self-adjoint operator [65]—a fact that is in agreement with Pauli’s theorem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The origin of this non-self-adjointness can be under- stood as a result of the singularity at p = 0 in the mo- mentum representation, ˆtAB → (iℏm/2)(p−2 − 2p−1∂p) [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nevertheless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' although the (generalized) eigenfunc- tions of ˆtAB are not orthogonal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' they constitute an over- complete set and provide a POVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' which are used to define the arrival-time distribution as follows [63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 65]: ΠSTD(t|x=L)= 1 2πℏ � α=± ����� � ∞ −∞ dp θ(αp) � |p| m ˜ψt(p)e i ℏ Lp ����� 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (6) where θ(·) is Heaviside’s step function and ˜ψt(p) is the wave function in the momentum representation which could be obtained from the initial wave function ˜ψ0(p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' as ˜ψt(p) = ˜ψ0(p) exp � − itp2/2mℏ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The distribution ΠSTD and its generalization in the presence of interaction po- tential have been referred to as the ”standard arrival- time distribution” by some authors [16, 66–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, Grot, Rovelli, and Tate treated the singularity of (5) by symmetric regularization and obtained equation (6) via the standard Born rule [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The generalizations of equa- tions (5) and (6) in the presence of interaction potential have been investigated in various works [16, 31, 70–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using these developments, it has been shown that the non-self-adjointness of the free arrival time operator can also be lifted by spatial confinement [71, 76], and the above arrival time distribution could be derived from the limit of the arrival time distribution in a confining box as the length of the box increases to infinity [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Fur- thermore, recently, the distribution (6) is derived from a space-time-symmetric extension of non-relativistic quan- tum mechanics [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 3 The three-dimensional generalization of (6) is derived by Kijowski’s [10] via an axiomatic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The as- sumed axioms are implied by the principle of the prob- ability theory, the mathematical structure of standard quantum mechanics, and the Galilei invariance [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Based on these axioms, Kijowski constructed the follow- ing arrival time distribution for a free particle that passes through a two-dimensional plane S as ΠSTD(t|x ∈ S) = 1 2πℏ � α=± � R2d2p∥ × ����� � ∞ −∞ dp⊥ θ(αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n) � |p⊥| m ˜ψt(p)e i ℏ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='p⊥ ����� 2 , (7) where p⊥ ≡(p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' n)n and p∥ ≡ p − p⊥ are perpendicular and parallel components of p relative to S respectively, and n is the outward normal of plane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In fact, he first proves the above expression for the wave functions whose supports lie in the positive (or negative) amounts of p⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Then he uniquely derives the following self-adjoint variant of the (three-dimensional version of) Aharonov- Bohm arrival time operator, by demanding that the time operator be self-adjoint and leads to (7) for these special cases via the Born rule [10, 78]: ˆtL = sgn(ˆp⊥) � mLˆp−1 ⊥ − m 2 (ˆp−1 ⊥ ˆx⊥ + ˆx⊥ˆp−1 ⊥ ) � , (8) where ˆx⊥ ≡ ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n and L (≡ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n) represent the distance between the detection surface and the origin [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Fi- nally, for an arbitrary wave function, the equation (7) could be derived from this self-adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' More- over, considering this time operator, besides the com- ponents of the position operator in the detection plane, ˆx∥ ≡ ˆx − (ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n)n, Kijowski obtains the following expres- sion as the joint position-time distribution on the detec- tion screen via the Born rule [78]: PSTD(x, t|x∈S) = � α=± |ψα S (x, t)|2, (9) in which ψ± S (x, t) is the wave function on the basics of joint eigenstates of the operators ˆtL and ˆx∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Explicitly ψ± S (x, t) = 1 (2πℏ)3/2 � d3p θ(±p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n) � |p⊥| m ˜ψt(p)e i ℏ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (10) Note that, the arrival time distribution (7) could be re- produced by taking the integral of (9) over the whole of the screen plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The joint space-time probability distri- bution (9), and its generalization for the particles with arbitary spin, have been also derived by Werner in an- other axiomatic manner [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, it is easy to see that the results (7) and (9) can be obtained from a reg- ularized version of the (three-dimensional generalization of) Aharonov-Bohm time operator, which is the same as the procedure used by Grot, Rovelli and Tate in one- dimensional cases [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Quantum flux and Bohmian approach Inspiring by classical intuition, another proper candi- date for screen observables is the perpendicular compo- nent of the quantum probability current to the screen surface, J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n, where J(x, t) = − ℏ m Im [ψ∗ t (x)∇ψt(x)] , (11) and n is the outward normal to the screen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This pro- posal is applicable for a particle in a generic external potential and a generic screen surface, not necessarily an infinite plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' There are several attempts to derive this proposal in various approaches, such as Bohmian mechanics for the scattering case in [79], decoherent his- tories approach in [80] as an approximation, or in [81] as an exact formula using the concept of extended probabil- ities, and so on [45, 46, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Howover, even if the wave function contains only momentum in the same direction as n, the J(x, t) · n could be negative due to the back- flow effect [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This property is incompatible with the standard notion of probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nevertheless, this prob- lem could be treated from the Bohmian point of view: Using Bohmian trajectories, it can be shown that the positive and negative values of J(x, t) · n correspond to the particles that reach the point x at S in the same di- rection of n or the opposite direction of it, respectively [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, through the Bohmian mechanics in one-dimension, Leavens demonstrates that the time distribution of arrival to x=L from both sides could be obtained from the absolute form of probability flux as [35, 85] ΠQF(t|x=L) = |J(L, t)| � dt |J(L, t)|, (12) which is free from the aforementioned problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The three-dimensional justification of J(x, t) · n as an operational formulation of the arrival time model has been made in [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Also, the generalization of (12) for arrival to the surface S is given by [7, 13, 16, 86] ΠQF(t|x∈S) = � S dS|J(x, t)·n| � dt � S dS|J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='n|, (13) with dS =n·dS the magnitude of the surface element dS which is directed outward at x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To illustrate (13) and to generalize it to the case of joint arrival distri- bution, we can use the Bohmian point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this theory, each particle has a specific trajectory, depending on the initial position, and so the rate of passing par- ticles through an area element dS centered at x ∈ S, in the time interval between t and t + dt, is proportional to ρt(x)|v(x, t)·dS|dt, where v(x, t)=J(x, t)/|ψt(x)|2 is the Bohmian velocity of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Hence, using quantum equilibrium condition [87, 88], ρt(x) = |ψt(x)|2, and ac- complishing normalization, the joint arrival distribution could be represented by the absolute value of the current density as 4 PQF(x, t|x∈S) = |J(x, t)·n| � dt � S dS|J(x, t)·n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) Now, by integrating (14) over all x ∈ S, we arrive at the three-dimensional arrival time distribution (13) for the screen surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' It should be noted that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) is not necessarily followed for an ensemble of classical par- ticles because a positive or negative current at a space- time point, (x, t), can in general have contributions from all the particles arriving to x at t from any direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, since the Bohmian velocity field is single- valued, the particle trajectories cannot intersect each other at any point of space-time and so only a single tra- jectory contributes to the current density J(x, t) at the particular space-time point (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, this fact implies that when v(x, t) · n>0 we can say that the tra- jectory and consequently the particle has passed through the screen from the inside and vice versa for v(x, t)·n<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Hence, one can define the joint probability distribution for the time of arrival to each side of S as P± QF(x, t|x∈S) = J±(x, t)·n � dt � S dS J±(x, t)·n, (15) where J±(x, t) = ± θ(±J·n) J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, note that there may be some trajectories which cross S more than once—and we have multi-crossing trajectories (see the typical Bohmian trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The course of the above inference to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) was in such a manner that multi-crossing trajectories could contribute several times (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, one could assume the detection surface as a barrier that does not allow the crossed particle to return inside (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this case, it is suggested to use the truncated current defined as ˜J(x, t) := �J(x, t) if (x, t) is a first exit through S 0 otherwise (16) where (x, t) is a first exit event through the boundary surface S, if the trajectory passing through x at time t leaves inside S at this time, for the first time since t = 0 [13, 79, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The limiting condition in (16), imposes that the joint probability distribution based on it should be computed numerically using trajectories: ˜PQF(x, t|x∈S) = ˜J(x, t)·n � dt � S dS ˜J(x, t)·n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (17) Of course, the detection screen is not always a barrier- like surface (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (b)), and one could assume that there is a point-like detector that lets the multi-crossing trajectories to contribute to the distribution and we can use (14) in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Horizontal screen Vertical screen Ly Lx x y s o FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Schematic double-slit experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The cen- ter of two slits is considered as the coordinate origin, and the vertical and horizontal screens are placed at x = Lx and y = Ly, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The dashed black line shows a typical Bohmian trajectory that arrives at the horizontal screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A suitable single-particle detector, in addition to particle arrival position, can record the arrival time using a proper clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' “INTRINSIC” SCREEN OBSERVABLE IN TWO-SLIT EXPERIMENT In this section, we study the discussed proposals in the previous section for the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We com- pare the results of these proposals in the cases of vertical and horizontal screens (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1), and also in different detection schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The main motivation for the study of the horizontal screen is the non-classical particles’ mo- tions along the y-direction, in the Bohmian perspective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' see a typical Bohmian trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This behav- ior is due to changing the sign of the probability cur- rent’s component in the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This behavior does not occur for x-component of J and consequently for the Bohmian motion of a particle along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1, the setup contains two identical slits at y = ±s, and screens are placed at x = Lx and y=Ly correspond to the vertical and horizontal screens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To avoid the mathematical complexity of Fresnel diffraction at the sharp-edge slits, it is supposed that the slits have soft edges that generate waves hav- ing identical Gaussian profiles in the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' So, for each slit, we can take the wave function as an uncorre- lated two-dimensional Gaussian wave packet, which in each dimension has the form ψ(i) G (x, t) = (2πs2 t)- 1 4 exp � (x − x(i) 0 − uxt)2 4σ0st � × exp � i ℏmux(x − x(i) 0 − uxt 2 ) � (i = 1, 2), (18) with m the particle’s mass, σ0 the initial dispersion, ux 5 S S S (a) (b) (c) First-arrival Second-arrival Third-arrival FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Different schemes of particle detection on the screen surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the Bohmian point of view, particles could have a recursive motion on surface S and cross it more than once (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' see the trajectory that plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Assuming different detector types, one can prob variant possible observables on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In panel (a) a conceivable particle trajectory is depicted, which crosses S three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this panel, a movable point-like detector is placed on S, which can survey the whole screen and detect particles that arrive only from one side, while in panel (b) a two-sided point detector is placed on S, which can move along it and detect particles that arrive from up and down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, one can assume there is (c) an array of side-by-side detectors covering the entire screen surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The last configuration blocks the trajectory and does not allow the crossed particle to return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this scheme, we only detect first-arrivals from one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the wave packet’s velocity, x(i) 0 the initial position of wave packet or in other words the location of i-th slit, and st = σ0(1 + iℏt/(2mσ2 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore, when the particle passes through the slits, we have the total wave function as ψ(x, y, t) = 1 √ 2[ψ(1) G (x, t)ψ(1) G (y, t) + ψ(2) G (x, t)ψ(2) G (y, t)], (19) where superscripts (1) and (2) correspond to upper and lower slits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This form of Gaussian superpo- sition state is commonly used in the literature [7, 90–93] and is feasible to implement by quantum technologies be- cause such a state could be produced and controlled read- ily [94, 95], even without using slits [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this paper, we have chosen the metastable helium atom, with mass m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='64 × 10−27 kg, as the interfering particle, and the parameters as s = 10 µm, σx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='04 µm, σy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 µm, ux = 3 m/s, and uy = 0 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These values are feasible according to the performed experiments [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, the meta-stable helium atom could be detected with high efficiency because of its large internal energy [52, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Vertical screen The arrival time distribution for the vertical screen placed at different distances from the two-slit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see this distribution is the same for all methods, and their average arrival time is close to the corresponding quantity in classical uniform motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To calculate the mean time of arrival to the screen, we use the arrival time distribution of each method presented in sec II, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (3), (7) and (13), and we have ¯tS = � ∞ 0 dt Π(t|x∈S) t, (20) as the mean arrival time at the surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Furthermore, we can compute the average arrival time to each point on the screen using the joint probability distribution as ¯tx = � ∞ 0 dt P(x, t|x∈S) t � ∞ 0 dt P(x, t|x∈S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21) This observable is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 4-b for a vertical screen placed at Lx = 300 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Apparently, the results of the standard and quantum flux methods are the same and similar to one that resulted in [7] by Nelson’s mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nevertheless, they are different from the semiclassical ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, when the interference pattern is calculated by either method, we see that their predicted cumulative position distributions do not differ much from the others (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 4-a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This observable can be calculated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='by using the joint distribution as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='▲▲▲▲▲▲▲▲ ' metadata={'source': 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+page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='130 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='100 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='Averaged arrival time (ms) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='P(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='y (mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 Semiclassical Standard Quantum flux FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (a) The cumulative arrival position distribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (22), for the vertical screen at Lx = 300 mm, and (b) the average arrival time at each point of the screen, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' P(x|x∈S)= � ∞ 0 dt P(x, t|x∈S) � ∞ 0 dt � S dS P(x, t|x∈S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (22) As mentioned, it should be noted that, |ψt(x)|2 is just the conditional position probability density at the specific time t, not the position-time joint probability density and so the accumulated interference pattern, P(x|x∈S), is not given by � dt|ψt(x)|2 [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Horizontal screen In this section, we are going to compare the mentioned proposals in the double-slit setup with a horizontal de- tection screen (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 5, the arrival time distributions at the screen are plotted for some horizontal screens which are located at Ly =15, 20, 25, and 30 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this figure, solid-black, dashed-green, and dash-dotted-blue curves represent the distributions ΠST D, ΠQF and ΠSC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Also, the vertical lines show the average time of arrival to the screen, ¯tS, associated with these arrival time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' From this figure, one can see that, although the averages al- most coincide, the distributions are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, as expected, when the screen’s distance from the center of the two slits Ly decreases, the difference between dis- tributions increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Most of these differences occur in the early times, which are associated with the particles that arrive at the S in the near field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Furthermore, we observe that the ΠSC behaves quite differently from ΠQF and ΠST D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The distributions ΠQF and ΠST D are more or less in agreement, however, for the screen that is lo- cated at Ly =15 µm, a significant difference between the standard and quantum flux distributions occurs around t≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='7 Π(t | y = Ly) t (ms) Ly = 30 µm Ly = 25 µm Ly = 20 µm Ly = 15 µm Semiclassical Standard Quantum flux FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Arrival time distributions of particles that arrive on the horizontal screen at four different distances from the center of two slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The vertical lines show the average arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To have a more comprehensive insight, we can look at the joint spatiotemporal arrival distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this figure, joint distributions, PSC, PSTD and PQF are plotted in three panels, for the horizontal screen surface located at Ly = 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These density plots clearly visualize differences between the mentioned arrival dis- tribution proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In these plots, we can see separated fringes with different shapes, which this fact imply that the particles arrive at the screen in some detached space-time regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the insets, one can see that the shapes of these regions are different for each proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the joint density of the semiclassical approximation 7 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6-a), fringes are well-separated, while the standard distribution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6-b) exhibits more continuity in its fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, in the pattern of the quantum flux proposal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6-c) there are grooves between every two fringes which is due to changing the sign of J(x, t) · n in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In all panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6, the duration of “temporal no-arrival windows” between every two typical fringes variate in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 ms which has a spatial extension of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3 to 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These space-time scales are utterly amenable empirically by current technologies [53, 96], which could be used to test these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 t (ms) t (ms) t (ms) x (mm) Quantum flux Standard Semiclassical (c) (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Density plots of joint arrival position-time distribu- tions for particles that arrive at the horizontal screen of the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panels (a), (b), and (c) represent PSC, PSTD and PQF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Insets: Magnified contour plots of the joint distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The average time of arrival to each point of the screen and cumulative position interference pattern could be cal- culated as in the vertical screen case by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(a)-(b), these two quantities are shown for the horizontal screen which is placed at y = 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In contrast to the vertical screen, the cumulative position distribution of the semiclassical approximation is entirely separate from the two other proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The cumulative position distribution resulting from standard and quan- tum flux approaches have obvious differences from each other, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(b), the aver- age arrival times are the same for all three methods at first and begin to deviate from each other at x ≈ 5 mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' then again, these curves converge to each other at x≈25 mm, approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The maximum deviation between the standard and quantum flux average arrival time oc- curs at x≈19 mm, which is quite in the far-field regime— the width of the initial wave function is ∼ O(10−3)mm which is smaller than 19 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Therefore one can suggest the average arrival time in the gray region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(b) as a practical target for comparing these approaches experi- mentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To this end, we study arrival time distributions at some points of this region as local arrival distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The arrival time distribution conditioned at a specific point x on the screen can be obtained as follow Πx(t|x∈S) = P(x, t|x∈S) � ∞ 0 dt P(x, t|x∈S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (23) Using the associated joint distribution of each proposal, we have plotted Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(c)-(f) that show Πx(t|x ∈ S) at the positions x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, on the screen placed at Ly = 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The broken black curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7 (c)-(f), resulting from the quantum flux proposal, against the smooth curves of the other two methods could be understood as the result of the changing the signa- ture of the y-component of the probability current: Note that, quantum flux distribution is given by the absolute value of the probability current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The origin of distinc- tions between the local average arrival times is more per- ceptible from these local arrival distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In princi- ple, these distributions could be probed using fast and high-resolution single-atom detectors [53, 97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In partic- ular, the delay-line detector that is recently developed by Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' [51] seems suitable for our purpose: It has the capability to resolve single-atom detection events temporally with 220 ps and spatially with 177µm at rates of several 106 events per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We estimate by a numerical investigation that these lo- cal arrival distributions could be well reconstructed from about 104 number of detection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As an example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7, the histograms associated with the probability densities of the panel (f) are plotted in panel (g), using 104 numerical random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' It is easy to estimate that the recording of 104 particle detection events can de- termine the local average arrival time with a statistical error of about 10−2ms, while the differences between local average arrival times of various proposals are almost big- ger than 10−1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using cumulative position distribution, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='90 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='08 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 x=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm x=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm Πx(t|x ∈ S) t (ms) ∆N (g) (f) (e) (d) (c) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1 0 1 0 600 Semiclassical Standard Quantum flux 0 5 10 15 20 25 0 2 4 6 8 3 4 5 6 7 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 5 12 14 16 18 20 22 24 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='005 Averaged arrival time (ms) x (mm) P(x) (b) (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The space-time arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panel (a) represents the average time of arrival at each point of the screen, ¯tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panel (b) represents the cumulative position probability den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The panels (c)-(f) show the local arrival time probability densities, Πx(t|x∈S), at the at the points x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm on the screen, which are chosen from the gray region in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The vertical lines in these panels represent the average arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Panel (g) is Histograms associated with probability densities of panel (f), which are generated by 104 numerical random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 7(b), one can estimate that, if the total number of particles that arrived at the screen is about 108, we have about 104 particles around x = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, in the spacial interval (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Using recent progress in laser cool- ing and magneto-optical trapping [97], the preparation of a coherent ensemble of metastable helium atoms with this number of particles is quite achievable [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' One might be inclined to think that the difference be- tween the quantum flux and standard average arrival times is just due to changing the signature of J(x, t) · n, but in the following, we show that even without the con- tribution of the negative part of J(x, t)·n, these proposals are significantly distinguishable: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Detection schemes As we mentioned in section II C, according to the Bohmian deterministic point of view, there are several possible schemes to detect arrived particles, especially for the horizontal screen surface which we have recursive motions on it (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' One can assume that the horizontal screen is swept with a point-like detector that surveys all arrived particles at the surface S, which we call spot-detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this scheme, one option is to use a unilateral detector to detect arrived particles at the top or bottom of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this case, the positive and negative parts of the quantum probability current have respectively corresponded to particles that arrive at the top or bottom of S (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (a)), and we must use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (15) to calculate the screen observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Addi- tionally, we can choose a bilateral detector (or two uni- lateral detectors) that prob all particles that arrive from both sides of S, along the time with several repeats of the experiment (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In these circum- stances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' spot-detection scheme), there is no barrier in front of the particles before they reach the point of detection and we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (14) to obtain the screen observables as in the two previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As we have already shown in section II C, whether the particles arrive from the top or bottom of S, the abso- lute value of the quantum probability current yield the trajectories’ density and consequently give the joint dis- tribution of the total arrival at each point of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This fact is the case for the standard method, as well, how- ever, there is a subtle difference between the two propos- als in the spot-detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' When we talk about 9 0 5 10 15 20 25 5 20 0 2 4 6 8 0 5 10 15 20 25 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='30 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 x (mm) t (ms) First arrivals Second arrivals Third arrivals All arrivals First arrivals Quantum flux FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The space-time Bohmian arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The interior curves in the central figure are the averaged times of arrival obtained by different detection schemes: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The Left and top plots are marginal arrival time distributions and marginal arrival position distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The scatter plot is generated using 2 × 106 Bohmian trajectories, and the black, blue, and green points of the scatter plot represent the first, second, and third arrivals of Bohmian particles to the screen, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The inset is a zoom-in of the dashed rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' the spot-detection in the Bohmian approach, it would be considered the possibility of multi-crossing and the distribution includes all-arrivals at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Although, in the standard method there is an interpretation for ψ+ S (x, t) and ψ− S (x, t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (10), which relates them to the par- ticles arrive at S in a direction which is the same or op- posite with the direction of outward normal of the screen n, respectively [10, 64], nevertheless, since there are no defined paths in this approach, it is obscure whether it counts only the first-arrivals to each side of the screen or includes recursive movements of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Alternatively, along with the spot-detection scheme, it could be assumed that there is a continuous flat barrier in front of the particle’s paths as the detection surface or screen surface that does not allow particles to cross this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Depending on the screen’s length and posi- tion, there are several possibilities for the detection pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In each case, a specific number of particle paths contribute to the distribution of arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the simplest case, the screen blocks all the trajectories that reach the horizontal surface S, and we only detect the first-arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In such a setup, we can no longer use the quantum flux method to represent Bohmian trajectories’ first encounter with the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' hence, the screen ob- servables must be obtained by numerical analysis, due to the definition of truncated current as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (16) and its corresponding joint distribution, ˜PQF(x, t|x∈S), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' By computing the Bohmian trajectories, we can find positions and times of the first-arrivals to the screen, and consequently calculate the arrival time distribution which mathematically could be defined as ˜ΠQF(t|x∈S) = � S ˜PQF(x, t|x∈S)dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (24) Also, other observable quantities such as the cumulative spatial distribution and averaged arrival time over the detection surface could be defined and calculated numer- ically in a similar way—by substituting ˜PQF(x, t|x ∈ S) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Furthermore, we can complete the computations to find the second and third encounters to the surface (regardless of the barrier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8, we show our numerical results of Bohmian trajectories simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The background scatter plot is the position and time of arrivals of 2 × 106 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this plot, the second and third arrivals are shown in blue and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Here, it is more clear why 10 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 x=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm x=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm Π(t| x, y) t (ms) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 First arrivals Quantum flux All arrivals FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Arrival time distribution at the horizontal screen po- sitions x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, and Ly = 15 µm, which are in the gray region of Fig (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The width of sampling in each point is about δx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='25 mm, and 108 Bohmian trajec- tories are simulated to obtain these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' we interpret the grooves of the quantum flux density plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 6 (c)) as a result of the multi-crossing of Bohmian trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The three middle graphs are the average time of the first and all-arrivals, which are simulation re- sults of 108 trajectories, and are compared by the quan- tum flux method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As expected, the average time of all- arrivals fits on the quantum flux curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, the av- erage time of first-arrivals deviates from all-arrivals in the area discussed in the previous section (between x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm and x = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To scrutinize the deviation zone of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8 (the gray re- gion), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 9 is drawn to show the arrival time distribu- tions of screen positions x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see, at the first recursive points of quantum flux distribution, the first-arrival distributions raise down to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This implies that in the presence of a barrier- like screen, there would be a big temporal gap between arrived particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These gaps could be investigated as a result of the non-intersection property of Bohmian tra- jectories that cause a unilateral motion of particles along the direction of the probability current field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' SCREEN BACK-EFFECT In principle, the presence of the detector could mod- ify the wave function evolution, before the particle detec- tion, which is called detector back-effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' To have a more thorough investigation of detection statistics, we should consider this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Howsoever, due to the measurement problem and the quantum Zeno effect [9], a complete in- vestigation of the detector effects is problematic at the fundamental level, and it is less obvious how to model an ideal detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Nonetheless, some phenomenological non-equivalent models are proposed, such as the gener- alized Feynman path integral approach in the presence of absorbing boundary [12, 37–39], Schr¨odinger equation with a complex potential [44], Schr¨odinger equation with absorbing (or complex Robin) boundary condition [40– 44], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The results of these approaches are not the same, and a detailed study of the differences is an in- teresting topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this section, we provide a brief review of the absorbing boundary rule (ABR) and path-Integral with absorbing boundary (PAB) models, then we com- pare them in the double-slit setup with the horizontal screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Absorbing Boundary Rule Among the above-mentioned phenomenological mod- els, the absorbing boundary condition approach has the most compatibility with Bohmian mechanics [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The application of absorbing boundary condition in arrival time problem was first proposed by Werner [40], and re- cently it is re-derived and generalized by Tumulka and others using various methods [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Especially, it is re- cently shown that in a suitable (non-obvious) limit, the imaginary potential approach yields the distribution of detection time and position in agreement with the ab- sorbing boundary rule [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' According to this rule, the particle wave function ψ evolves according to the free Schr¨odinger equation, while the presence of a detection screen is modeled by imposing the following boundary conditions on the Detection screen, x ∈ S, n · ∇ψ = iκψ, (25) where κ>0 is a constant characterizing the type of detec- tor, in which ℏκ/m represents the momentum that the detector is most sensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This boundary condition ensures that waves with wave number κ are completely absorbed while waves with other wave numbers are partly absorbed and partly reflected [41, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In the absorbing boundary rule, the joint spatiotemporal distribution of the detection event is given by quantum flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Consider- ing (25), this distribution reads PABR(t, x|x∈S) = |ψABC|2 � dt � S dS|ψABC|2 , (26) where ψABC represent the solution of the free Schr¨odinger equation satisfying the aforementioned absorbing bound- ary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This distribution can be understood in terms of Bohmian trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The Bohmian particle equation of motion, ˙X = (ℏ/m)Im [∇ψABC/ψABC], to- gether with the boundary condition (25), imply that tra- jectories can cross the boundary S only outwards and so there are no multi-crossing trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' If it is assumed 11 that the detector clicks when and where the Bohmian particle reaches S, the probability distribution of detec- tion events is given by (26), because the initial distribu- tion of the Bohmian particle is |ψABC(x, 0)|2 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Path-Integral with Absorbing Boundary In several papers [12, 37–39], Marchuwka and Schuss develop an interesting method to calculate the detec- tion effect of absorbing surface using the Feynman path integral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' They postulate a separation princi- ple for the wave function in which we could consider the (bounded wave function) as a sum of two parts, ψ(x, t) = ψ1(x, t) + ψ2(x, t), such that ψ1(x, t) corre- sponds to the survival part of the wave which is orthogo- nal to ψ2(x, t) at a time t and evolve independently [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' So, we can obtain the probability of survival of the parti- cle, denoted S(t), which is the probability of the particle not being absorbed by the time t, as � D d3x|ψ1(x, t)|2, where the integral is over the domain D, outside the ab- sorbing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' By discretizing the path integral in a time interval [0, t] and eliminating the trajectories that, in each time interval [t′, t′+∆t′] for all t′ < t, are reached to the absorbing surface S, the survival and consequently absorbing probability would be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Based on this analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' we could define a unidirectional probability cur- rent into the surface as d dt[1−S(t)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' which yields a normal component of the multidimensional probability current density at any point on S as J(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' t)·n= λℏ mπ |n·∇ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' t)|2 × exp � − λℏ mπ � t 0 dt′ � S dS|n·∇ψ(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' t′)|2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' (27) where dS = n · dS is the magnitude of the surface ele- ment dS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' n is the unit outer normal to the absorbing surface S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' and λ is a proportionality factor with the di- mension of length [37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Also, ψ(x, t) is the solution of Schr¨odinger equation bounded and normalized in the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, the normal component J(x, t)·n is supposed to be the probability density for observing the particle at the point x on the screen at time t [12, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Screen back-effect in two-slit experiment In order to complete the investigations carried out in section III, we are going to study the screen back- effect in the double-slit experiment with a horizontal screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, we compare the arrival distri- butions which are resulted from the absorbing bound- ary rule (ABR), path-Integral with absorbing boundary (PAB), and Bohmian truncated current (BTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We continue with the same initial conditions as in sec- tion III, and choose κ = 1 µm−1 for ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='00 t (ms) t (ms) t (ms) x (mm) Bohmian truncated current Path-Integral with Absorbing Boundary Absorbing Boundary Rule (c) (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Density plots of joint probability distributions of position and time (screen observable) for the horizontal screen placed at y = 15 µm in the double-slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These densities are calculated by the three methods which take the screen effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' κ leads to the maximum absorption probability—which is almost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4—for the chosen initial wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, for a more meaningful comparison, we consider λ = 1 µm in the PAB method, which leads to the same absorption probability as ABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The resulting joint ar- rival time-position distributions of the three methods are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' As one can see, the distributions of the ABR and PAB methods—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=', panels (a) and (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10—have more compatibility with each other than the result of the BTC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, there are dif- ferences between them which are more obvious in the zoomed areas.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='x=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm x=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4 mm x=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm Π(t| x, y) t (ms) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='5 ABR PAB BTC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Arrival time distribution at the horizontal screen po- sitions x = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='2 mm, and Ly = 15 µm, which are calculated for the three methods which take the screen effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' formly distributed than of the PAB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The empty areas between the fringes of the panel (c) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 10 are due to the elimination of the recursive trajectories—or in other words, are due to the elimination of second and third arrivals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' For a more detailed comparison, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 11 the spa- tial and temporal marginal distributions are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In addition, the associated local average arrival times are compared in the central panel of this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The PAB method leads to significant discrepancies in marginal dis- tributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The maximum difference is about 40% that occurs around x≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content='8 mm, which seems testable clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In contrast to the previous results on intrinsic distri- butions, in which the difference between average arrival times was significant, there is a good agreement in this observable for the ABR and PAB methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' However, there is a significant difference between the average ar- rival time in these two methods and BTC around x = 6 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 12, the local arrival time distributions at some points on the screen are plotted, which show simi- lar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' SUMMARY AND DISCUSSION When and where does the wave function collapse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' How one can model a detector in quantum theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' These are the questions that we investigated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' We tried to show that there is no agreed answer for these ques- tions, even for the double-slit experiment that has in it the heart of quantum mechanics [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This is a practical encounter with the measurement problem [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this regard, we numerically investigated and compared the main proposed answers to these questions for a double- slit setup with a horizontal detection screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' It is shown that these proposals lead to experimentally distinguish- able predictions, thanks to the current single-atom de- tection technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' In this work, we suggest the meta-stable helium atom as a proper coherent source of the matter wave, however, other sources may lead to some practical improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' For example, using heavier condensate atoms can lead to more clear discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Moreover, it is worth not- ing that although the experiment with photons may have some practical advantages, there are more complications in its theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' This is partially because of the relativistic localization-causality problem [101–104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' The theoretical investigation of a proposed experiment for photons would be an interesting extension of the present work, which has been left for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' ACKNOWLEDGMENTS We sincerely thank Mohammad Hossein Barati for carefully reviewing the manuscript, and Sheldon Gold- stein for his helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' 13 [1] As shown by Pauli [105], if the Hamiltonian spectrum is discrete or has a lower bound, then there is no self-adjoint time operator canonically conjugate to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Zimmermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Mishra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Doran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Gordon, and A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Johnson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Emary, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' See, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Grif- fiths, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Jones, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Farrer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Ritchie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' Pepper, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE0T4oBgHgl3EQfxAG2/content/2301.02641v1.pdf'} +page_content=' B.' 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