# Double-slit experiment remastered

Siddhant Das<EMAIL_ADDRESS>Dirk-André Deckert
<EMAIL_ADDRESS>Leopold Kellers<EMAIL_ADDRESS>Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr.
39, D-80333 München, Germany  Ward Struyve<EMAIL_ADDRESS>Instituut
voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B–3001 Leuven,
Belgium  Centrum voor Logica en Filosofie van de Wetenschappen, KU Leuven,
Kardinaal Mercierplein 2, B-3000 Leuven, Belgium

###### Abstract

Time-of-flight measurements of helium atoms in a double-slit experiment
reported in [C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997)]
are compared with the arrival times of Bohmian trajectories. This is the first
qualitative comparison of a quantum mechanical arrival-time calculation with
observation at the single-particle level, particularly noteworthy given the
absence of a consensus in extracting time-of-flight predictions from quantum
theory. We further explore a challenging double-slit experiment in which one
of the slits is shut in flight.

Diffraction and interference phenomena occupy a prominent place in the
phenomenology of quantum physics, Young’s double-slit experiment (DSE) being
an archetypal example which, according to Feynman “has in it the heart of
quantum mechanics” Feynman  _et al._ (2011). Quantum physicists have discussed
the DSE and augments thereof, e.g., the which-way Wootters and Zurek (1979);
Tan and Walls (1993), or the delayed-choice DSE Wheeler (1978), with varying
degrees of rigour. Several notable realisations employing _single_ electrons
Bach  _et al._ (2013); Tonomura  _et al._ (1989), neutrons Zeilinger  _et al._
(1988); Gähler and Zeilinger (1991), atoms Carnal and Mlynek (1991);
Kurtsiefer  _et al._ (1997), and even macro-molecules Brand  _et al._ (2021)
have been performed. However, an _essential_ aspect of this paradigmatic
experiment amenable to observation, _viz._ , the arrival (or detection) times
of the particle at the detector, remains elusive.

While it is practically impossible to predict exactly where or when the
particle will scintillate on the screen in a given experimental run, _every_
discussion of the DSE is at least concerned with the interference fringes
formed by the accumulation of single-particle impact positions depicted, e.g.,
in Fig. 1 (a). What tends to be overlooked, however, is that each imprinted
position with screen coordinates $(x,y)$ is associated with a definite, albeit
random, detection-time $t_{f}$—the time-of-flight (ToF) of the particle from
the source to the screen. Figure 1 (b) gives the distribution of $t_{f}$ and
the $x$-screen-coordinate of the very same detection events underlying Fig. 1
(a). Therefore, specifying the distribution of $t_{f}$ (or, better, the joint
probability distribution of $x$, $y$, and $t_{f}$) is indispensable for a
satisfactory account of the DSE, or any other scattering experiment for that
matter. In this letter, we try to fill this gap between theory and practice.

Figure 1: (a) Double-slit interference pattern observed on a screen after many
single-atom detection events. (b) Joint distribution of the detection-time (or
time-of-flight) $t_{f}$ and the $x$-screen-coordinate of the detection events.
Impact positions of atoms arriving in the $t_{f}$ range indicated by the
dashed lines in (b) were accumulated to produce (a). A fraction of very fast
atoms emanating from the source cast a shadow of the slits at the bottom.
(This figure is reproduced from Kurtsiefer and Mlynek (1996) with permission.)

The reader might be inclined to think that the distribution in Fig. 1 (b)
corresponds to $\smash{\int\\!dy\kern 1.00006pt|\psi(x,y,d,t_{f})|^{2}}$,
where $\psi(x,y,z,t)$ is the particle’s wave function at time $t$ and
$\smash{z=d}$ the detection plane. But this is incorrect because
$|\psi(x,y,z,t)|^{2}$ is, according to Born’s rule, the distribution of
_positions_ detected at a _specified time_ $t$, whereas what is needed is a
distribution of _detection times_ (and positions detected at such times) at a
_specified surface_.111The conjectured quantity is not even normalizable in
simple examples. Therefore, it cannot be a legitimate probability density.

The measurements underlying Fig. 1 were performed by Kurtsiefer, Pfau, and
Mlynek (hereafter KPM) Kurtsiefer  _et al._ (1997); Kurtsiefer and Mlynek
(1996); Pfau and Kurtsiefer (1997) in the 1990s using metastable (i.e.,
electronically excited) helium atoms (see below for further details). As yet,
the measured arrival times and impact positions have not been compared with
theory. This lacuna is not accidental. The description of the arrival times by
itself is a long-standing problem with various proposed (disparate) solutions
Muga and Leavens (2000); Muga  _et al._ (1998); Das and Dürr . That there is
no recognized observable for time in quantum mechanics, in contrast to
position, is only partly responsible for this. While the adequacy of many
existing proposals is questionable Mielnik and Torres-Vega (2005); Leavens
(2002); *Mugareply; *Leavensreply; Das and Nöth (2021); Das and Struyve
(2021), so far none of the suggestions have been benchmarked against
experiment. To our knowledge, this work is the first comparison of this sort.

On the other hand, KPM themselves were mainly interested in the measurement
(or, rather, reconstruction) of the Wigner function from ToF data Leibfried
_et al._ (1998) theorised in earlier work Janicke and Wilkens (1995). As just
indicated, standard quantum mechanics does not provide an unambiguous notion
of arrival-time of a particle.222It has even been suggested that “time of
arrival cannot be precisely defined and measured in quantum mechanics”
Aharonov  _et al._ (1998). Therefore, any tomographic reconstruction of the
Wigner function from ToF measurements is somewhat questionable. We will not
address this issue in this letter.

Instead, we will present numerical evidence showing that the statistics of
impact positions _and_ arrival times obtained by KPM can be reproduced by a
pragmatic application of Bohmian mechanics. The conceptual resources provided
by this theory, _viz._ , point particles moving on definite trajectories, make
it ideal for handling ToF experiments (as has long been recognized Daumer  _et
al._ (1997); Leavens (1998a, b); Grübl and Rheinberger (2002); Cushing
(1995)). Bohmian trajectories are widely used in the literature for addressing
arrival and tunneling-time problems, see Kazemi and Hosseinzadeh (2022); Xie
_et al._ (2022); Leavens (1996, 1993); Douguet and Bartschat (2018); Das and
Dürr (2019); *Exotic; Oriols  _et al._ (1996); Mousavi and Golshani (2008a);
Mousavi (2015); Nogami  _et al._ (2000); Ruggenthaler  _et al._ (2005); Pan
and Home (2012); Home and Majumdar (1999); *HomeCP; Mousavi and Golshani
(2008b); Demir (2022); Field (2022). Subsequently, we consider a challenging
variant of the DSE, dubbed dynamic DSE, in which one of the slits is shut in
flight. Finally, we conclude with some implications for the foundations of
quantum mechanics.

## Elements of Bohmian mechanics.

Bohmian mechanics (also called the de Broglie-Bohm theory or pilot-wave
theory) is a theory about point-particles moving along trajectories that
grounds the formalism and the predictions of standard quantum mechanics Bohm
and Hiley (1993); Holland (1995); Dürr and Teufel (2009); Dürr  _et al._
(2004). For a single-particle with position $\mathbf{R}$ and mass $m$, the
dynamics is given by the guidance equation

$\dot{\mathbf{R}}(t)=\frac{\hbar}{m}\kern
1.00006pt\imaginary\\!\left[\frac{\boldsymbol{\nabla}\psi\big{(}\mathbf{R}(t),t\big{)}}{\psi\big{(}\mathbf{R}(t),t\big{)}}\right],$
(1)

where $\psi(\mathbf{r},t)$ is the wave function satisfying Schrödinger’s
equation with an external potential $V(\mathbf{r},t)$. The wave function never
collapses in this theory. Here, the “wave/particle duality” of quantum
mechanics is resolved in a trivial way: both a wave _and_ a particle are
present.

The motion of the particle is deterministic, i.e., given its initial position
$\mathbf{R}(0)$ and wave function $\psi(\mathbf{r},0)$, it has a unique
trajectory. However, in different experimental runs the initial positions are
typically random, with distribution given by $|\psi(\mathbf{r},0)|^{2}$—the
quantum equilibrium distribution. For this reason the arrival times of a
particle in different runs of a scattering experiment are random, as are the
locations where it strikes the detector surface.

## Accounting for the DSE.

Bohmian trajectories for the DSE shown in Fig. 2 (b) are well-known; they
feature invariably in expositions of Bohmian mechanics. DSE trajectories were
first presented by Philippidis _et al._ in the late 1970s Philippidis  _et
al._ (1979) and have been reproduced various times using different methods
Philippidis  _et al._ (1982); Sanz  _et al._ (2002); Holland and Philippidis
(2003); Gondran and Gondran (2005). (Here, they are produced for the first
time without simplifying assumptions on the level of the wave function.) See
also Kocsis  _et al._ (2011) for a weak measurement of average trajectories in
a DSE, which resemble those of Fig. 2 (b).

The trajectories provide an almost self-explanatory account of how the
interference pattern builds up on a distant screen one particle at a time. In
particular, they show “how the motion of a particle, passing through just one
of two holes in [a] screen, could be influenced by waves propagating through
both holes. And, so influenced that the particle does not go where the waves
cancel out, but is attracted to where they cooperate” (Bell, 2004, p. 191). It
follows that one could have particle trajectories and still account for
interference experiments, claims to the contrary notwithstanding.333For
instance, “it is clear that [the DSE] can in no way be reconciled with the
idea that electrons move in paths. … In quantum mechanics there is no such
concept as the path of a particle.” (Landau and Lifshitz, 1977, p. 2), or that
“many ideas have been concocted to try to explain the curve for $P_{12}$ [the
interference pattern] in terms of individual electrons going around in
complicated ways through the holes. None of them has succeeded.” (Feynman  _et
al._ , 2011, Sec. 1.5).

Figure 2: (a) Schematic drawing of the double-slit potential (3) for
$\smash{f=1}$. The $y$-axis points out of the page, along which the potential
is translationally invariant. (b) $xz$-projections of a collection of Bohmian
trajectories for the DSE with initial conditions (dots) sampled randomly from
the $|\psi(\mathbf{r},0)|^{2}$-distribution. Each curve tracks a particle that
eventually strikes the detection plane $\smash{z=d}$ at the bottom. The
following potential barrier and wave packet parameters were assumed:
$\smash{V_{0}=532.5}$, $\smash{\sigma_{b}=0.075}$, $\smash{\sigma_{s}=0.25}$,
$\smash{k_{x}=x_{0}=0}$, $\smash{k_{z}=10.5}$, $\smash{z_{0}=2}$, and
$\smash{\sigma_{x}=3\kern 1.00006pt\sigma_{z}=0.75}$ in the units where
$\smash{\hbar=m=\Delta=1}$. Inset: Characteristic non-Newtonian meandering of
the Bohmian trajectories in the near-field.

For producing the Bohmian trajectories in Fig. 2 (b), we solved Schrödinger’s
equation numerically with initial wave function

$\psi(\mathbf{r},0)=\frac{e^{i(k_{x}x+k_{z}z)}}{\sqrt{\sigma_{x}\kern
1.00006pt\sigma_{y}\kern 1.00006pt\sigma_{z}}}\kern
1.00006ptG\\!\left(\frac{x+x_{0}}{\sigma_{x}}\right)G\\!\left(\frac{y}{\sigma_{y}}\right)G\\!\left(\frac{z+z_{0}}{\sigma_{z}}\right),$
(2)

$\smash{G(\xi)=\pi^{-1/4}\exp(-\,\xi^{2}/2)}$—a normalized Gaussian _wave
packet_ centred at $(-\kern 1.00006ptx_{0},0,-\kern 1.00006ptz_{0})$ behind
the slits. The screen containing the slits was modelled by the Gaussian
potential barrier

$\displaystyle V=V_{0}\kern
1.00006ptG\\!\left(\frac{z}{\sigma_{b}}\right)\\!\left[1-S\\!\left(\frac{x-\Delta}{\sigma_{s}}\right)-f\kern
1.00006ptS\\!\left(\frac{x+\Delta}{\sigma_{s}}\right)\right]$ (3)

featuring two rectangular apertures for $\smash{f=1}$,444For the dynamic DSE
problem considered below, $f$ is varied in time as per Eq. (5). as shown in
Fig. 2 (a). Here,
$\smash{S(\xi)=1\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}(1+\xi^{16})}$, $2\Delta$, $\sigma_{s}$, and
$\sigma_{b}$ respectively denote the aperture function, inter-slit separation,
slit-width, and the thickness of the barrier. Throughout the present work,
masses, lengths and times are expressed in units of $m$, $\Delta$ and
$\smash{m\Delta^{2}\kern-1.00006pt\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}\hbar}$, respectively. (This is equivalent to
setting $\smash{\hbar=m=\Delta=1}$.) The Bohmian trajectories were computed by
numerically integrating the guidance equation (1) for $260$ initial positions
randomly sampled from the $|\psi(\mathbf{r},0)|^{2}$-distribution. About 25%
of the sampled trajectories made it through the slits giving rise to detection
events. The rest (not shown in Fig. 2 (b)) were back-scattered from the
potential barrier.

## From pictures to predictions.

The preceding discussion suggests that an individual detection event is
triggered close to the location and time-of-impact of a Bohmian trajectory. In
view of this, the detection-time on a specified surface $\mathcal{D}$ (such as
the plane $\smash{z=d}$) would correspond to the arrival (first passage or
hitting) time of a Bohmian trajectory555One considers the first passage time
because in general a trajectory might cross the surface $\mathcal{D}$ more
than once Das and Dürr (2019); *Exotic. However, this does not happen in the
present setting.:

$t_{f}(\mathbf{r}_{0})=\text{inf}\kern
1.00006pt\big{\\{}\,t:\mathbf{R}(t)\in\mathcal{D}\text{ and
}\mathbf{R}(0)=\mathbf{r}_{0}\,\big{\\}},$ (4)

$\mathbf{R}(t_{f})$ being the concomitant detected position on
$\mathcal{D}$,666The infimum in (4) ensures that the arrival-time of
trajectories not intercepting $\mathcal{D}$, e.g., those back-scattered at the
plane containing the slits giving rise to non-detection events, is infinity,
since $\text{inf}\kern 1.00006pt\emptyset:=\infty$. as a function of the
random initial condition $\mathbf{r}_{0}$—quantities that have no respective
counterparts in standard quantum theory. In fact, in standard quantum theory,
the detection event $\smash{(x,y,t_{f})}$ must be defined by the observation-
induced collapse of the wave function, whose time of occurrence is ambiguous.

Barring special circumstances, the distribution of this detection event
$\big{(}\mathbf{R}(t_{f}),t_{f}\big{)}$ is only numerically accessible. For
instance, in the far-field or scattering regime, where backflow is absent,
this distribution typically reduces to $j_{\perp}(\mathbf{r},t_{f})$ Kreidl
_et al._ (2003); Daumer  _et al._ (1997); *DDGZ96, (Das and Nöth, 2021, p.
7)—the component of the quantum flux (or probability current) density
orthogonal to the surface $\mathcal{D}$. Evidently, integrating this (joint)
distribution over all $t_{f}$ yields the (marginal) distribution of the
detected positions alone, e.g., Fig. 1 (a). Thus arrival-time considerations
are indispensable for correctly reproducing the usual interference pattern.
(Note, by the way, that the quantum flux statistics is _not_ determined by a
POVM Vona  _et al._ (2013), as would be required by the standard quantum
formalism.)

Various suggested arrival-time distributions are defined with reference to a
detector model, typically characterised by new phenomenological parameters,
e.g., Włodarz (2002); Marchewka and Schuss (1998); *PI2; *PI3; Dubey  _et al._
(2021); *ABC; Blanchard and Jadczyk (1998); Jurman and Nikolić (2021); Galapon
_et al._ (2005); *CTOA; *CTOAerror. In our case, a very compelling qualitative
agreement with the experimental data is obtained by disregarding the micro-
physics of the detector altogether, and without introducing any free
parameters. While a more detector-aware Bohmian analysis could be
envisaged777In certain situations the Bohmian degrees of freedom of the
detector particles must be taken into account exclusively, e.g., in the which-
way and delayed choice DSEs Tastevin and Laloë (2018); Wiseman (1998)(Bell,
2004, Ch. 14), or in Wigner’s friend scenarios Lazarovici and Hubert (2019).,
this is easier said than done, and our results indicate that this may not even
be necessary.

## Specifics of the KPM experiment.

In Kurtsiefer  _et al._ (1997), the metastable helium atoms (denoted He^*) are
sourced from a gas-discharge tube by firing a $15\,\mu\text{s}$ pulse of
electric current. A few spurious fast-moving atoms along with slower atoms
with velocities ranging between $\smash{1-3\times 10^{3}\,\text{ms}^{-1}}$
(Pfau and Kurtsiefer, 1997, Fig. 4) are produced by the source. The ejected
atoms pass through a skimmer and are collimated by a
$\smash{5\,\mu\text{m}}$-wide slit. Further downstream, they encounter a
microfabricated double-slit structure of inter-slit separation
$\smash{2\Delta=8\,\mu\text{m}}$ and slit-width $\sigma_{s}=1\,\mu\text{m}$,
eventually striking the detector plate at $d$.

He^* atoms eject secondary electrons upon hitting the conductive surface of
the detector—a process assumed to take place on a time scale of
$10^{-12}\,\text{s}$. These electrons are carefully imaged onto a single-
electron detection unit based on a chevron assembly of multichannel plates
Kurtsiefer and Mlynek (1996). The spatial (temporal) resolution achieved with
this detector is on the order of $500\,\text{nm}$ ($100\,\text{ns}$).

For three different slit-screen separations $d$, KPM recorded the detection
events $(x,y,t_{f})$ of individual atoms by repeatedly firing the source.
Their observations are shown in Fig. 3 (left panel). Notice that the fast
atoms produce a “geometrical shadow” of the slits at the bottom of each figure
(Kurtsiefer  _et al._ , 1997, p. 151).

## A numerical treatment of the KPM experiment.

Assuming an initial Gaussian wave packet (2),888A nonzero
$x_{\scalebox{0.75}{0}}$ and $k_{x}$ was assumed in order to reflect the
manifest left-right asymmetry of the experimental plots, perhaps the result of
some misalignment. we solved Schrödinger’s equation with the double-slit
potential (3). The wave function then determines the particle velocity in the
guidance equation (1). Numerically integrating Eq. (1) for $5\times 10^{4}$
initial conditions randomly sampled from the $|\psi(\mathbf{r},0)|^{2}$
distribution, we recorded the impact positions
$\big{(}X(t_{f}),Y(t_{f})\big{)}$ and arrival times $t_{f}$ of Bohmian
trajectories arriving at $\smash{z=d}$.999Since separated wave functions imply
independence of orthogonal motions in Bohmian mechanics (Holland, 1995, Sec.
3.3.5.), details of the motion parallel to the $y$-axis neither affect the ToF
$t_{f}$ nor the $x$-screen-coordinate $X(t_{f})$. Histograms of obtained
$t_{f}$ and $X(t_{f})$ are presented in Fig. 3 (right panel).

Figure 3: Joint distributions of arrival times and $x$-screen coordinates for
different slit-screen separations. Left panel: Experimental data for He^*
atoms with $\smash{d=148\,\text{mm}}$ (a), $\smash{d=248\,\text{mm}}$ (b), and
$\smash{d=1,950\,\text{mm}}$ (c) (reproduced from Kurtsiefer  _et al._ (1997)
with permission). Here, $\smash{t_{d}=(d/L)\,t_{f}}$ is a “scaled ToF”, where
$L$ is the source-to-screen separation. Right panel: Numerical data generated
from Bohmian trajectory impacts for $\smash{V_{0}=665.7}$,
$\smash{\sigma_{b}=0.125}$, $\smash{\sigma_{s}=0.5}$, $\smash{k_{x}=-\,0.5}$,
$\smash{k_{z}=15.7}$, $\smash{z_{0}=4}$, $\smash{x_{0}=-\,0.13}$,
$\smash{\sigma_{x}=2.25}$, $\smash{\sigma_{z}=0.25}$, with $\smash{d=0.9}$
(d), $\smash{d=9}$ (e), and $\smash{d=15}$ (f), in the units where
$\smash{\hbar=m=\Delta=1}$.

We made a few assumptions in order to enable numerics: (A) The large spatial
extent of KPM’s experimental set-up prevents a straightforward solution of the
time-dependent Schrödinger equation, therefore we chose wave packet and
potential barrier parameters that have similar relative values in our units.
(B) Our particles start directly behind the double slits, rather than at the
source. Since $k_{z}$ is very large, all particles have practically the same
transit-time from the source to the slits, which can therefore be ignored in
computing the total $t_{f}$ without affecting the shape of the arrival-time
distribution significantly. It is the double-slit potential (3) which
entangles the motions in the $x$ and $z$ directions correlating the particle
arrival-times with the $x$-screen coordinate of their impact positions,
captured in Fig. 2. (C) Finally, we have employed the same initial wave
function for each run of the DSE, even though the discharge tube yields a
statistical mixture of wave packets of different longitudinal velocities.
Lacking additional details of the KPM discharge source, we chose this simpler
treatment.

Despite these simplifications, we do find a good qualitative agreement between
the raw data reported by KPM and our numerical calculations in Fig. 3. This
indicates that the obtained statistics are generic and are not too sensitive
to the aspects we have glossed over. Arguably, a better characterisation of
the source in the analysis or more accurate numerical simulations can only
make the agreement better. Also, our results further underscore the point
(discussed earlier) that a microscopic account of the detector for predicting
the measured arrival times is certainly unnecessary for producing results
comparable in accuracy to the usual interference pattern discussed in
textbooks.

## Dynamic DSE.

In what follows, we set

$f=\frac{1}{2}\kern 1.00006pt\bigg{\\{}\kern
1.00006pt1+\tanh\kern-1.00006pt\big{[}\gamma\kern
1.00006pt(t_{c}-t)\big{]}\kern 1.00006pt\bigg{\\}},$ (5)

in (3), which causes the slit centred at $\smash{x=-\kern 1.00006pt\Delta}$ to
gradually close in time. In particular, for $\smash{t\gg(\ll)\,t_{c}}$,
$\smash{f\approx 0\,(1)}$ implying a completely closed (open) slit. (A
dynamically controlled DSE is well-amenable to present-day technology Bach
_et al._ (2013).) Scatter plots of $\big{(}X(t_{f}),t_{f}\big{)}$ for this
dynamic DSE are plotted in Fig. 4 for two values of $\gamma$; a large (small)
$\gamma$ implies a fast (slow) closing of the slit. The closing time $t_{c}$
is chosen close to the time at which the peak of the wave packet strikes
$\smash{z=0}$, i.e.,
$\smash{z_{0}\raisebox{-0.86108pt}{\leavevmode\resizebox{}{6.88889pt}{\kern
1.00006pt/\kern 1.00006pt}}k_{z}}$. Evidently, a smaller number of
trajectories pass through the closing slit. Furthermore, the interference
fringes become less pronounced for $\smash{t_{f}\gg t_{c}}$, as would be
expected for single-slit diffraction.

Figure 4: Scatter plot of $\smash{\approx 10^{4}}$ Bohmian trajectories
arriving at $\smash{d=10}$ over time for the dynamic DSE, where the slit
centred at $\smash{x=-\kern 1.00006pt\Delta}$ ($\smash{=-\kern 1.00006pt1}$ in
our units) is closed in flight around time $\smash{t_{c}=0.25}$: (a) Fast
switching $\smash{\gamma=100}$, (b) Slow switching $\smash{\gamma=20}$. Data
points for trajectories passing through the closing (open) slit are rendered
in blue (beige). Potential barrier and initial wave packet parameters are same
as Fig. 2.

## Concluding remarks.

The computation of the probability distribution of arrival or detection times
of quantum particles is one of the last areas where experts disagree about
what quantum mechanics should predict. While a cornucopia of different ToF
distributions exists in the literature, one rarely finds a discussion of the
joint distribution of impact positions _and_ arrival times studied in this
work (and measured in the KPM experiment). In this letter, we have presented a
Bohmian calculation of such distributions, obtaining a compelling qualitative
agreement with the KPM measurements.

While Bohmian mechanics, owing to the well-defined concepts of point-particles
and trajectories embedded in it, is readily suited for explaining the results
of such ToF experiments,101010The same can be said for Nelson’s stochastic
mechanics—a trajectory containing quantum theory à la Bohmian mechanics but
based on Brownian paths instead of smooth ones given by Eq. (1)—which can also
describe arrival times in the DSE in a straightforward manner Nitta and Kudo
(2008). it is as yet unclear how alternative quantum theories, e.g.,
spontaneous/objective collapse and many-worlds theories, among others, would
deal with the same in a practicable way.

It must be emphasized that the Bohmian trajectories serve not only to anchor
one’s intuition, but also to provide theoretical resources for interpreting
laboratory ToF experiments that otherwise canonically rely on Newtonian
concepts. The applicability of such concepts is often questionable, e.g., in
discussing tunneling situations111111This is reflected in the interpretation
of the latest “tunneling-time” experiments featuring electrons Sainadh  _et
al._ (2020) and atoms Ramos  _et al._ (2020) that reached exactly opposing
conclusions, _viz._ , zero and non-zero tunneling-time, respectively. (see
also Das and Struyve (2021)). Given that ToF measurements are routinely
performed to measure/reconstruct physical quantities of interest, a first
principles underpinning of ToF measurements becomes all the more necessary.

To make progress with this question, we recommend that ToF resolved DSEs and
dynamic variants thereof, as described in this work, should be performed with
present-day technology to achieve a _quantitative_ experiment-to-theory
comparison in the future. A particular improvement would be better control of
the initial wave function of the particle, e.g., sourcing the particle from an
ion trap post-cooling Stopp  _et al._ (2022), or using field-emission-tip
electron wave packets à la Jones  _et al._ (2016), both of which offer better
initial wave packet control compared to a gas-discharge source.

## Acknowledgements.

J. Dziewior, C. Kurtsiefer, T. Maudlin, M. Mukherjee, S. Goldstein, and H.
Ulbricht are thanked for fruitful discussions, H. Weinfurter for suggesting
the dynamic DSE, and J. M. Wilkes for editorial inputs. L.K. and D.D.
acknowledge funding from the Elite Network of Bavaria, through the Junior
Research Group “Interaction Between Light and Matter”. W.S. is supported by
the Research Foundation Flanders (Fonds Wetenschappelijk Onderzoek, FWO),
Grant No. G066918N. We dedicate this work to the memory of Detlef Dürr.

## References

  * Feynman  _et al._ (2011) R. Feynman, R. Leighton, and M. Sands,  _The Feynman Lectures on Physics, Quantum Mechanics Vol. III_ (Basic Books, 2011).
  * Wootters and Zurek (1979) W. K. Wootters and W. H. Zurek, Phys. Rev. D 19, 473 (1979).
  * Tan and Walls (1993) S. M. Tan and D. F. Walls, Phys. Rev. A 47, 4663 (1993).
  * Wheeler (1978) J. A. Wheeler, in  _Mathematical Foundations of Quantum Theory_, edited by A. Marlow (Academic Press, 1978) pp. 9–48.
  * Bach  _et al._ (2013) R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, New J. Phys. 15, 033018 (2013).
  * Tonomura  _et al._ (1989) A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, Am. J. Phys. 57, 117 (1989).
  * Zeilinger  _et al._ (1988) A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe, Rev. Mod. Phys. 60, 1067 (1988).
  * Gähler and Zeilinger (1991) R. Gähler and A. Zeilinger, Am. J. Phys. 59, 316 (1991).
  * Carnal and Mlynek (1991) O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991).
  * Kurtsiefer  _et al._ (1997) C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature 386, 150 (1997).
  * Brand  _et al._ (2021) C. Brand, S. Troyer, C. Knobloch, O. Cheshnovsky, and M. Arndt, Am. J. Phys. 89, 1132 (2021).
  * Kurtsiefer and Mlynek (1996) C. Kurtsiefer and J. Mlynek, Appl. Phys. B 64, 85 (1996).
  * Pfau and Kurtsiefer (1997) T. Pfau and C. Kurtsiefer, J. Mod. Opt. 44, 2551 (1997).
  * Muga and Leavens (2000) J. G. Muga and C. R. Leavens, Phys. Rep. 338, 353 (2000).
  * Muga  _et al._ (1998) J. Muga, R. Sala, and J. Palao, Superlattices and Microstructures 23, 833 (1998).
  * (16) S. Das and D. Dürr, “Benchmarking quantum arrival time distributions using a back-wall,” in preparation.
  * Mielnik and Torres-Vega (2005) B. Mielnik and G. Torres-Vega, Concepts of Physics. II, 81 (2005).
  * Leavens (2002) C. R. Leavens, Phys. Lett. A 303, 154 (2002).
  * Egusquiza  _et al._ (2003) I. Egusquiza, J. Muga, B. Navarro, and A. Ruschhaupt, Phys. Lett. A 313, 498 (2003).
  * Leavens (2005) C. R. Leavens, Phys. Lett. A 345, 251 (2005).
  * Das and Nöth (2021) S. Das and M. Nöth, Proc. R. Soc. A. 477, 20210101 (2021).
  * Das and Struyve (2021) S. Das and W. Struyve, Phys. Rev. A 104, 042214 (2021).
  * Leibfried  _et al._ (1998) D. Leibfried, T. Pfau, and C. Monroe, Phys. Today 51, 22 (1998).
  * Janicke and Wilkens (1995) U. Janicke and M. Wilkens, J. Mod. Opt. 42, 2183 (1995).
  * Aharonov  _et al._ (1998) Y. Aharonov, J. Oppenheim, S. Popescu, B. Reznik, and W. G. Unruh, Phys. Rev. A 57, 4130 (1998).
  * Daumer  _et al._ (1997) M. Daumer, D. Dürr, S. Goldstein, and N. Zanghì, J. Stat. Phys. 88, 967 (1997).
  * Leavens (1998a) C. Leavens, Phys. Rev. A 58, 840 (1998a).
  * Leavens (1998b) C. R. Leavens, Superlattices and Microstructures 23, 795 (1998b).
  * Grübl and Rheinberger (2002) G. Grübl and K. Rheinberger, J. Phys. A: Math. Gen. 35, 2907 (2002).
  * Cushing (1995) J. T. Cushing, Found. Phys. 25, 269 (1995).
  * Kazemi and Hosseinzadeh (2022) M. Kazemi and V. Hosseinzadeh, arXiv:2208.01325 (2022).
  * Xie  _et al._ (2022) W. Xie, M. Li, Y. Zhou, and P. Lu, Phys. Rev. A 105, 013119 (2022).
  * Leavens (1996) C. R. Leavens, “The “tunneling-time problem” for electrons,” in  _Bohmian Mechanics and Quantum Theory: An Appraisal_, edited by J. T. Cushing, A. Fine, and S. Goldstein (Springer Netherlands, Dordrecht, 1996) pp. 111–129.
  * Leavens (1993) C. Leavens, Phys. Lett. A 178, 27 (1993).
  * Douguet and Bartschat (2018) N. Douguet and K. Bartschat, Phys. Rev. A 97, 013402 (2018).
  * Das and Dürr (2019) S. Das and D. Dürr, Sci. Rep. 9, 2242 (2019).
  * Das  _et al._ (2019) S. Das, M. Nöth, and D. Dürr, Phys. Rev. A 99, 052124 (2019).
  * Oriols  _et al._ (1996) X. Oriols, F. Martín, and J. Suñé, Phys. Rev. A 54, 2594 (1996).
  * Mousavi and Golshani (2008a) S. V. Mousavi and M. Golshani, Phys. Scr. 78, 035007 (2008a).
  * Mousavi (2015) S. V. Mousavi, Phys. Scr. 90, 095001 (2015).
  * Nogami  _et al._ (2000) Y. Nogami, F. Toyama, and W. van Dijk, Phys. Lett. A 270, 279 (2000).
  * Ruggenthaler  _et al._ (2005) M. Ruggenthaler, G. Grübl, and S. Kreidl, J. Phys. A: Math. Gen. 38, 8445 (2005).
  * Pan and Home (2012) A. K. Pan and D. Home, Int. J. Theor. Phys 51, 374–389 (2012).
  * Home and Majumdar (1999) D. Home and A. S. Majumdar, Found. Phys. 29, 721 (1999).
  * Majumdar and Home (2002) A. S. Majumdar and D. Home, Phys. Lett. A 296, 176 (2002).
  * Mousavi and Golshani (2008b) S. V. Mousavi and M. Golshani, J. Phys. A: Math. Theor. 41, 375304 (2008b).
  * Demir (2022) D. Demir, Phys. Rev. A 106, 022215 (2022).
  * Field (2022) G. E. Field, Euro. Jnl. Phil. Sci. 12 (2022).
  * Bohm and Hiley (1993) D. Bohm and B. J. Hiley,  _The Undivided Universe: An Ontological Interpretation of Quantum Theory_ (Routledge, London and New York, 1993).
  * Holland (1995) P. R. Holland,  _The quantum theory of motion: an account of the de Broglie-Bohm causal interpretation of quantum mechanics_ (Cambridge university press, 1995).
  * Dürr and Teufel (2009) D. Dürr and S. Teufel,  _Bohmian Mechanics: The Physics and Mathematics of Quantum Theory_ (Springer-Verlag, Berlin, 2009).
  * Dürr  _et al._ (2004) D. Dürr, S. Goldstein, and N. ZanghÌ, J. Stat. Phys. 116, 959 (2004).
  * Philippidis  _et al._ (1979) C. Philippidis, C. Dewdney, and B. Hiley, IL Nuov Cim B 52, 15 (1979).
  * Philippidis  _et al._ (1982) C. Philippidis, D. Bohm, and R. D. Kaye, IL Nuov Cim B 71, 75 (1982).
  * Sanz  _et al._ (2002) A. S. Sanz, F. Borondo, and S. Miret-Artés, J. Phys.: Condens. Matter 14, 6109 (2002).
  * Holland and Philippidis (2003) P. Holland and C. Philippidis, Phys. Rev. A 67, 062105 (2003).
  * Gondran and Gondran (2005) M. Gondran and A. Gondran, Am. J. Phys. 73, 507 (2005).
  * Kocsis  _et al._ (2011) S. Kocsis _et al._ , Science 332, 1170 (2011).
  * Bell (2004) J. S. Bell,  _Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy_ (Cambridge University Press, 2004).
  * Landau and Lifshitz (1977) L. D. Landau and E. M. Lifshitz,  _Quantum Mechanics: Nonrelativistic theory_ , 3rd ed., Course of Theoretical Physics, Vol. 3 (Pergamon Press, Moscow, 1977).
  * Kreidl  _et al._ (2003) S. Kreidl, G. Grübl, and H. G. Embacher, J. Phys. A: Math. Gen. 36, 8851 (2003).
  * Daumer  _et al._ (1996) M. Daumer, D. Dürr, S. Goldstein, and N. Zanghì, Lett. Math. Phys. 38, 103 (1996).
  * Vona  _et al._ (2013) N. Vona, G. Hinrichs, and D. Dürr, Phys. Rev. Lett. 111, 220404 (2013).
  * Włodarz (2002) J. J. Włodarz, Phys. Rev. A 65, 044103 (2002).
  * Marchewka and Schuss (1998) A. Marchewka and Z. Schuss, Phys. Lett. A 240, 177 (1998).
  * Marchewka and Schuss (2001) A. Marchewka and Z. Schuss, Phys. Rev. A 63, 032108 (2001).
  * Marchewka and Schuss (2002) A. Marchewka and Z. Schuss, Phys. Rev. A 65, 042112 (2002).
  * Dubey  _et al._ (2021) V. Dubey, C. Bernardin, and A. Dhar, Phys. Rev. A 103, 032221 (2021).
  * Tumulka (2022) R. Tumulka, Ann. Phys. 442, 168910 (2022).
  * Blanchard and Jadczyk (1998) P. Blanchard and A. Jadczyk, Int. J. Theor. Phys. 37, 227 (1998).
  * Jurman and Nikolić (2021) D. Jurman and H. Nikolić, Phys. Lett. A 396, 127247 (2021).
  * Galapon  _et al._ (2005) E. A. Galapon, R. Caballar, and R. Bahague, Phys. Rev. A 72, 062107 (2005).
  * Galapon  _et al._ (2004) E. A. Galapon, R. F. Caballar, and R. T. B. Jr, Phys. Rev. Lett. 93, 180406 (2004).
  * Galapon  _et al._ (2008) E. A. Galapon, R. F. Caballar, and R. T. B. Jr, Phys. Rev. Lett. 101, 169901 (2008).
  * Tastevin and Laloë (2018) G. Tastevin and F. Laloë, Eur. Phys. J. D 72, 033018 (2018).
  * Wiseman (1998) H. M. Wiseman, Phys. Rev. A 58, 1740 (1998).
  * Lazarovici and Hubert (2019) D. Lazarovici and M. Hubert, Sci Rep 9, 470 (2019).
  * Nitta and Kudo (2008) H. Nitta and T. Kudo, Phys. Rev. A 77, 014102 (2008).
  * Sainadh  _et al._ (2020) U. S. Sainadh, R. T. Sang, and I. V. Litvinyuk, J. Phys. Photonics 2, 042002 (2020).
  * Ramos  _et al._ (2020) R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature 583, 529 (2020).
  * Stopp  _et al._ (2022) F. Stopp, H. Lehec, and F. Schmidt-Kaler, Quantum Sci. Technol. (2022).
  * Jones  _et al._ (2016) E. Jones, M. Becker, J. Luiten, and H. Batelaan, Laser Photon Rev. 10, 214 (2016).