# On the experiment-friendly formulation of quantum backflow

Maximilien Barbier  Max-Planck-Institut für Physik komplexer Systeme,
Nöthnitzer Straße 38, D-01187 Dresden, Germany  Center for Nonlinear Phenomena
and Complex Systems, Université Libre de Bruxelles (ULB), Code Postal 231,
Campus Plaine, B-1050 Brussels, Belgium  Arseni Goussev  School of Mathematics
and Physics, University of Portsmouth, Portsmouth PO1 3HF, United Kingdom

###### Abstract

In its standard formulation, quantum backflow is a classically impossible
phenomenon in which a free quantum particle in a positive-momentum state
exhibits a negative probability current. Recently, Miller et al. [Quantum 5,
379 (2021)] have put forward a new, “experiment-friendly” formulation of
quantum backflow that aims at extending the notion of quantum backflow to
situations in which the particle’s state may have both positive and negative
momenta. Here, we investigate how the experiment-friendly formulation of
quantum backflow compares to the standard one when applied to a free particle
in a positive-momentum state. We show that the two formulations are not always
compatible. We further identify a parametric regime in which the two
formulations appear to be in qualitative agreement with one another.

##  1 Introduction

Quantum backflow (QB), as originally introduced in Ref. [1]111The existence of
quantum backflow was first pointed out by Allcock [2] and Kijowski [3], but
these were Bracken and Melloy [1] who carried out the first systematic
analysis of the phenomenon. In particular, they were the first to discuss it
for normalized states., refers to the classically forbidden fact that a free
particle may exhibit a negative probability current at a particular space-time
point even though it has, with certainty, a positive momentum. This original
formulation of QB is concerned with a nonrelativistic structureless quantum
particle of mass $m$ that follows a free one-dimensional motion along the $x$
axis. We will be denoting the state of the particle at time $t$ by
$\left\lvert\psi_{t}\right\rangle$ and its position representation by
$\psi_{t}(x)\equiv\langle x|\psi_{t}\rangle$. The probability current
$j_{t}(a)$ at a position $x=a$ is then given by

$\displaystyle
j_{t}(a)\equiv\frac{\hbar}{m}\operatorname{Im}\left[\psi_{t}^{*}(x)\frac{\partial}{\partial
x}\psi_{t}(x)\,\right]_{x=a}\,,$ (1)

where $\operatorname{Im}(z)$ and $z^{*}$ denote the imaginary part and complex
conjugate of a complex number $z$, respectively. If
$\left\lvert\psi_{t}\right\rangle$ is a positive-momentum state, then QB
occurs whenever [1]

$\displaystyle j_{t}(a)<0$ (2)

at some position $a$ and time $t$.

The phenomenon of QB has been investigated in a variety of different
scenarios, and many explicit examples of backflowing states have been
constructed, see e.g. Refs. [1, 4, 5, 6, 7]. Thus, QB has been considered for
a particle moving in the presence of linear [8] and short-range [9]
potentials. It has been extended to rotational motion [10, 11, 12], as well as
to relativistic [13, 14, 15] and many-particle systems [16]. The spatial
extent of QB has been addressed in Refs. [4, 17, 9]. QB is also known to bear
close relation to the arrival-time problem [2, 3, 18, 19, 20, 21, 22, 23, 24,
25] and some nonclassical aspects of the flow of probability in quantum
systems [26, 27, 28, 29, 7].

As of today, QB has never been observed experimentally. It has been argued
that QB can be observed in experiments with Bose-Einstein condensates [30,
31]. Recently, an experimental realization of an optical counterpart of QB has
been reported [32].

Of particular significance to the present study is Ref. [33] that addresses QB
for a particle, moving in a potential $V(x)$, whose state $\psi_{t}(x)$
contains a priori both positive and negative momenta. In order to treat this
scenario, the authors propose an alternative, “experiment-friendly” (EF)
definition of QB, which is based on replacing the right-hand side of Eq. (2)
by an integral involving the negative momenta. This new formulation of QB has
the advantage of being applicable in scattering situations. However, as we
show in this brief paper, unless used with care the EF criterion of Ref. [33]
may fail to identify QB in the standard case of a free particle with a
positive momentum.

Our paper is organized as follows. In Section 2, we provide a concise summary
of the EF formulation of QB put forward in Ref. [33]. In Section 3, we analyze
a concrete example of a positive-momentum state that is known to exhibit QB
according to the standard criterion, Eq. (2). We show that, for some parameter
values, the EF criterion of Ref. [33] can be violated for this particular
state. Then, in Section 4, we address the question of the maximal backflow,
computed in accordance with the EF formulation, that can be obtained for an
arbitrary positive-momentum state. We demonstrate that this maximal backflow
may become negligibly small in a certain parametric regime. Finally,
conclusions are drawn in Section 5.

##  2 Statement of the problem

A one-dimensional nonrelativistic structureless quantum particle can be
described by the wave function $\psi_{t}(x)$ in position space, or
alternatively by its Fourier transform $\widetilde{\psi}_{t}(p)$ in momentum
space. Both functions satisfy the normalization condition

$\displaystyle\int_{\mathbb{R}}dx\,\lvert\psi_{t}(x)\rvert^{2}=\int_{\mathbb{R}}dp\,\lvert\widetilde{\psi}_{t}(p)\rvert^{2}=1\,,$
(3)

so that $\lvert\psi_{t}(x)\rvert^{2}$ (respectively,
$\lvert\widetilde{\psi}_{t}(p)\rvert^{2}$) is interpreted as the probability
density of finding the particle at position $x$ (respectively, with momentum
$p$) at time $t$. Hereinafter, the Fourier transform $\widetilde{g}(p)$ of a
function $g(x)$ is taken to be

$\displaystyle\widetilde{g}(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{\mathbb{R}}dx\,e^{-ipx/\hbar}g(x)\,,$
(4a) with the inverse transform hence given by $\displaystyle
g(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{\mathbb{R}}dp\,e^{ixp/\hbar}\,\widetilde{g}(p)\,.$
(4b)

As is well known (see e.g. [34, 35]), the fact that the position and momentum
observables do not commute precludes the construction of any well-defined
quantum probability distribution in phase space. Instead, every quantum state
can be associated with infinitely many phase-space distributions, all of them
being functions of the phase-space variables $x$ and $p$, i.e., classical-like
commuting variables. None of these functions however simultaneously satisfies
all of the following three defining properties of probabilities, also known as
Kolmogorov’s axioms: positivity, normalizability, and additivity (see e.g.
[36]). This is the reason why quantum phase-space distributions are often
referred to as quasiprobability distributions. While all these distributions
embed the same physical information, their mathematical properties may
drastically differ from one another. Commonly used quasiprobability
distributions include the Wigner, Husimi, and Glauber-Sudarshan
representations.

In Ref. [33], the authors consider a particular class of phase-space
distributions $f_{t}(x,p)$ that are everywhere positive. This property, along
with the normalization condition

$\displaystyle\int_{\mathbb{R}^{2}}dxdp\,f_{t}(x,p)=1\,,$ (5)

allows one to assign a (quasi)probabilistic meaning to the distribution
$f_{t}$. The latter is defined by

$\displaystyle f_{t}(x,p)=\left\lvert W_{\psi_{t},\chi}(x,p)\right\rvert^{2}$
(6)

in terms of the so-called Wigner-Moyal transform [37]

$W_{\psi_{t},\chi}(x,p)\equiv\frac{1}{\sqrt{2\pi\hbar}}\\\\[5.69046pt]
\times\int_{\mathbb{R}}dy\,e^{-ipy/\hbar}\chi^{*}\left(y-\frac{x}{2}\right)\psi_{t}\left(y+\frac{x}{2}\right)\,.$
(7)

The function $\chi$ represents the precision function of a measurement
apparatus, and is normalized to unity222There is a slight difference between
the definition of the Wigner-Moyal transform in Eq. (7) and the way the
transform is defined in Eq. (2) of Ref. [33]: The prefactor in the right-hand
side of Eq. (7) is $1/\sqrt{2\pi\hbar}$, whereas it is $1/2\pi\hbar$ in Ref.
[33]. The expression in Ref. [33] is recovered from Eq. (7) by means of the
substitution $\chi(x)=\phi(x)/\sqrt{2\pi\hbar}$., i.e.

$\displaystyle\int_{\mathbb{R}}dx\left\lvert\chi(x)\right\rvert^{2}=1\,.$ (8)

It is easy to see from Eq. (4) that $W_{\psi_{t},\chi}$ can also be expressed
as an integral over momentum, namely

$W_{\psi_{t},\chi}(x,p)=\frac{1}{\sqrt{2\pi\hbar}}\\\\[5.69046pt]
\times\int_{\mathbb{R}}dp^{\prime}\,e^{ixp^{\prime}/\hbar}\,\widetilde{\chi}^{*}\left(p^{\prime}-\frac{p}{2}\right)\widetilde{\psi}_{t}\left(p^{\prime}+\frac{p}{2}\right)\,.$
(9)

The “experiment-friendly” (EF) definition of quantum backflow (QB) put forward
in Ref. [33] states that QB takes place at point $x=a$ and time $t$ if the
probability current $j_{t}(a)$ satisfies the inequality

$\displaystyle j_{t}(a)<\frac{1}{m}\int_{\mathbb{R}^{-}}dp\,pf_{t}(a,p)\,.$
(10a) Hereinafter we use the notation $\mathbb{R}^{-}$ (respectively,
$\mathbb{R}^{+}$) for the set of negative (respectively, positive) real
numbers. Note that the criterion (10a) can be alternatively written as
$\displaystyle J_{t}(a)<0$ (10b)

in terms of the quantity $J_{t}$ defined as

$\displaystyle J_{t}(a)\equiv
j_{t}(a)-\frac{1}{m}\int_{\mathbb{R}^{-}}dp\,pf_{t}(a,p)\,.$ (11)

The form of condition (10b) is reminiscent of the standard QB criterion (2),
with $J_{t}(a)$ playing the role of an effective backflow current at position
$a$ and time $t$.

Let us look into the structure of criterion (10) in more detail. First, note
that in view of (1) the left-hand side of Eq. (10a) depends solely on the
state $\psi_{t}$ of the system. However, it is clear from Eqs. (6) and (7)
that the right-hand side of Eq. (10a) depends on $\psi_{t}$ as well as on the
precision function $\chi$. The latter is by construction an arbitrary
function, independent of the state $\psi_{t}$. This means that by changing the
precision function $\chi$ one changes the right-hand side of Eq. (10a) while
leaving the left-hand side unchanged. Therefore, in principle, criterion (10)
could, for a given state $\psi_{t}$, be satisfied for one precision function
$\chi$ but violated for another. In fact, this is precisely what we
demonstrate in Section 3 below.

From here on, let us focus on a Gaussian precision function of the form

$\displaystyle\chi(x)=\frac{1}{\pi^{1/4}\sqrt{\sigma}}\,e^{-x^{2}/2\sigma^{2}}\,.$
(12a) The corresponding momentum representation is given by
$\displaystyle\widetilde{\chi}(p)=\frac{1}{\pi^{1/4}\sqrt{\widetilde{\sigma}}}\,e^{-p^{2}/2\widetilde{\sigma}^{2}}\,.$
(12b)

Here, the position- and momentum-space widths $\sigma$ and
$\widetilde{\sigma}$, respectively, are related via
$\sigma\widetilde{\sigma}=\hbar$. Our motivation for this choice of $\chi$ is
twofold. First, a Gaussian smoothing function is the most natural choice for
mimicking a finite precision of a measurement apparatus333For instance,
measurements are modeled by Gaussian quasiprojectors in [5]., and, as such, is
the precision function used in all examples of Ref. [33]. Second, it follows
from Eqs. (6) and (7) that the phase-space distribution $f_{t}$ obtained using
a Gaussian $\chi$ corresponds to the Husimi distribution (see e.g. Eq. (7.25)
in [34]), which is arguably the most commonly used nonnegative quantum phase-
space distribution function.

Our aim is to compare the EF criterion (10) against the standard definition of
QB, based on Eq. (2). The latter only applies to a free particle in a
positive-momentum state, i.e., a state described by the momentum wave function

$\displaystyle\widetilde{\psi}_{t}(p)=e^{-ip^{2}t/2m\hbar}\widetilde{\psi}_{0}(p)$
(13)

that vanishes identically for negative momenta,

$\displaystyle\widetilde{\psi}_{t}(p)=0\qquad\text{if}\qquad p<0\,.$ (14)

So, below we only consider wave functions of this form.

In the present paper, we argue that the usefulness of the EF definition of QB,
given by Eq. (10), is strongly dependent on the width $\widetilde{\sigma}$ of
the precision function. We present our argument in the following two sections.
Thus, in Section 3, we consider the case of a particle prepared in a
particular positive-momentum state $\widetilde{\psi}_{0}$ that satisfies the
standard criterion (2) of QB, and show that the EF criterion (10) gets
violated if $\widetilde{\sigma}$ exceeds a certain value. Then, in Section 4,
we address the problem of the maximal backflow probability transfer
$\Delta_{\max}$ achievable with positive-momentum states. On the one hand,
numerical analysis based on the standard backflow criterion (2) yields [1, 4,
38]

$\displaystyle\Delta_{\text{max}}^{(\text{BM})}\approx 0.0384517\,,$ (15)

a number commonly referred to as the Bracken-Melloy bound. On the other hand,
we demonstrate that $\Delta_{\max}$ inferred from the EF criterion (10) decays
with $\widetilde{\sigma}$ and can become arbitrarily small. This means that
the new criterion (10) may fail to identify QB if the precision function is
too broad in momentum space. We further quantify the range of
$\widetilde{\sigma}$ in which the EF definition of QB is consistent with the
standard one.

##  3 Explicit example

Let us consider the particular positive-momentum state that was introduced in
Ref. [1] as a concrete example of QB for normalized states. The state
corresponds to a fixed instant in time, taken to be $t=0$, and is given by

$\widetilde{\psi}_{0}(p)\equiv\left\\{\begin{array}[]{ll}0&,\quad
p<0\\\\[8.5359pt]
\frac{18p}{\sqrt{35\alpha^{3}}}\left(e^{-p/\alpha}-\frac{1}{6}\,e^{-p/2\alpha}\right)&,\quad
p>0\end{array}\right.$ (16a) where $\alpha$ is a positive constant that has
the dimension of a momentum. Note that the function $\widetilde{\psi}_{0}(p)$
is continuous at $p=0$. Substituting Eq. (16a) into Eq. (4b), one obtains the
corresponding position representation [1]
$\displaystyle\psi_{0}(x)=18\sqrt{\frac{\alpha\hbar^{3}}{70\pi}}\left[\frac{1}{(\hbar-i\alpha
x)^{2}}-\frac{2}{3}\frac{1}{(\hbar-2i\alpha x)^{2}}\right]\,.$ (16b)

The probability current $j_{0}(0)$ at $a=0$, calculated according to Eq. (1),
yields [1]

$\displaystyle j_{0}(0)=-\frac{36\alpha^{2}}{35\pi m\hbar}<0\,.$ (17)

The criterion (2) is thus clearly satisfied, meaning that the particular
positive-momentum state (16) is a backflowing state in the standard sense.

We now investigate the predictions of the EF criterion (10) for the state
(16). To this end, we first compute the corresponding phase-space distribution
$f_{0}(0,p)$ obtained from Eq. (6) for $x=0$ and $t=0$. In order to focus on
the role played by the width $\widetilde{\sigma}$ of the Gaussian precision
function, we use a system of units that is adapted to this particular state by
introducing the dimensionless momentum

$\displaystyle\eta\equiv\frac{p}{\alpha}\,.$ (18)

Then, combining the definitions (6) and (9) with Eqs. (12b), (16a) and (18),
we find

$\displaystyle
f_{0}(0,\alpha\eta)=\frac{162}{35\pi^{3/2}\hbar}\frac{I^{2}(\eta;s)}{s}\,,$
(19)

where $s$ is the dimensionless momentum width of the precision function,

$\displaystyle s\equiv\frac{\widetilde{\sigma}}{\alpha}\,,$ (20)

and $I$ is given by the integral

$\displaystyle
I(\eta;s)\equiv\int_{\mathbb{R}^{+}}d\eta^{\prime}\,\eta^{\prime}e^{-(\eta^{\prime}-\eta)^{2}/2s^{2}}\left(e^{-\eta^{\prime}}-\frac{e^{-\eta^{\prime}/2}}{6}\,\right).$
(21)

Alternatively, $I$ can be expressed as

$I(\eta;s)=se^{-\eta^{2}/2s^{2}}\left\\{\frac{5s}{6}\right.\\\
\left.-\sqrt{\frac{\pi}{2}}(s^{2}-\eta)e^{(s^{2}-\eta)^{2}/2s^{2}}\operatorname{erfc}\left(\frac{s^{2}-\eta}{\sqrt{2}s}\right)\right.\\\\[5.69046pt]
\left.+\sqrt{\frac{\pi}{2}}\frac{s^{2}-2\eta}{12}e^{(s^{2}-2\eta)^{2}/8s^{2}}\operatorname{erfc}\left(\frac{s^{2}-2\eta}{2\sqrt{2}s}\right)\right\\}$
(22)

with

$\displaystyle\operatorname{erfc}(z)=1-\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}dy\,\mathrm{e}^{-y^{2}}$
(23)

being the complementary error function.

Substituting Eqs. (17) and (19) into Eq. (11), we obtain the effective
backflow current:

$\displaystyle
J_{0}(0)=-\frac{18}{35\pi}\frac{\alpha^{2}}{m\hbar}\left[2+\frac{9}{\sqrt{\pi}s}\int_{\mathbb{R}^{-}}d\eta\,\eta
I^{2}(\eta;s)\right]\,.$ (24)

It is clear from this expression that the sign of $J_{0}(0)$ only depends on
the (dimensionless) width $s$ of the precision function. In particular, note
that the integral in the right-hand side of (24) is necessarily negative.

Figure 1: Scaled effective backflow current (solid blue curve), as given by
(24), as a function of the (dimensionless) width $s$ of the Gaussian precision
function.

We now evaluate $J_{0}(0)$ for various values of $s$ by numerically computing
the integral in Eq. (24). The results are shown in Fig. 1, where the (scaled)
effective backflow current is depicted by the solid blue curve. We see that
for small enough values of $s$ the current $J_{0}(0)$ takes negative values
(hashed green region). In this regime, the EF criterion (10b) predicts the
occurrence of QB and is in agreement with the standard backflow criterion (2).
However, for $s\gtrsim 5.6$ the current $J_{0}(0)$ takes positive values
(hashed red region). In this regime, condition (10) is no longer fulfilled,
implying that the EF criterion fails to identify QB.

This shows that the compatibility between the EF criterion (10) and the
standard one, Eq. (2), is sensitive to the momentum width $\widetilde{\sigma}$
of the precision function. While this conclusion was reached based on the
particular state (16), we now investigate the predictions of condition (10)
for general positive-momentum states.

##  4 Maximal backflow

We now consider the following question, originally posed in Ref. [1]: For a
free particle in a positive-momentum state and a given time interval $(0,T)$,
what is the maximal amount of probability $\Delta_{\max}$ that can possibly
cross the space point $x=0$ in the “wrong” direction? If one relies on the
standard definition of QB, Eq. (2), the answer to this question is given by
the Bracken-Melloy bound, Eq. (15). Here, we want to answer this question
adopting the new, EF definition of QB, Eq. (10), and compare the results with
the standard case.

To this end, we use the approach developed in Ref. [1]. The particle is
described by the wave function

$\displaystyle\psi_{t}(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{\mathbb{R}^{+}}dp\,e^{ixp/\hbar}\,e^{-ip^{2}t/2m\hbar}\widetilde{\psi}_{0}(p)\,.$
(25)

Substituting Eq. (25) into Eq. (1) and setting $a=0$, we obtain

$j_{t}(0)=\frac{1}{4\pi
m\hbar}\int_{\left(\mathbb{R}^{+}\right)^{2}}dpdp^{\prime}\,\widetilde{\psi}_{0}^{*}(p)\\\
\times(p+p^{\prime})e^{it\left(p^{2}-p^{\prime
2}\right)/2m\hbar}\widetilde{\psi}_{0}(p^{\prime})\,.$ (26)

Then, upon combining the definitions (6) and (9) with Eqs. (12b), (13), and
(14), we get

$\frac{1}{m}\int_{\mathbb{R}^{-}}dp\,pf_{t}(0,p)=\frac{1}{4\pi
m\hbar}\int_{\left(\mathbb{R}^{+}\right)^{2}}dpdp^{\prime}\,\widetilde{\psi}_{0}^{*}(p)\\\
\times U(p,p^{\prime};\widetilde{\sigma})e^{it\left(p^{2}-p^{\prime
2}\right)/2m\hbar}\widetilde{\psi}_{0}(p^{\prime})\,,$ (27)

where the function $U$ is defined as

$\displaystyle U(p,p^{\prime};\widetilde{\sigma})$
$\displaystyle\quad\equiv\frac{2}{\sqrt{\pi}\widetilde{\sigma}}\int_{\mathbb{R}^{-}}dp^{\prime\prime}\,p^{\prime\prime}e^{-[(p^{\prime\prime}-p)^{2}+(p^{\prime\prime}-p^{\prime})^{2}]/2\widetilde{\sigma}^{2}}\,.$
(28)

The function $U$ can also be expressed in terms of the complementary error
function (see Eq. (34) below). Substituting Eqs. (26) and (27) into Eq. (11)
yields the effective backflow current

$J_{t}(0)=\frac{1}{4\pi
m\hbar}\int_{\left(\mathbb{R}^{+}\right)^{2}}dpdp^{\prime}\,\widetilde{\psi}_{0}^{*}(p)\\\
\times\left[p+p^{\prime}-U(p,p^{\prime};\widetilde{\sigma})\right]e^{it\left(p^{2}-p^{\prime
2}\right)/2m\hbar}\widetilde{\psi}_{0}(p^{\prime})\,.$ (29)

The backflow probability transfer through $x=0$ over the time interval $0<t<T$
is given by

$\displaystyle\Delta\equiv-\int_{0}^{T}dt\,J_{t}(0)\,.$ (30)

The maximization of $\Delta$ is performed under the constraint that
$\widetilde{\psi}_{0}$ is normalized according to Eq. (3). Substituting Eq.
(29) into Eq. (30), evaluating the time integral, and using the method of
Lagrange multipliers, we find that the maximum of $\Delta$ (subject to the
normalization constraint) is given by the largest eigenvalue
$\Delta_{\text{max}}$ $(\equiv\sup\limits\lambda)$ of the following integral
eigenvalue problem:

$\int_{\mathbb{R}^{+}}dp^{\prime}\,\frac{i}{2\pi}\frac{p+p^{\prime}-U(p,p^{\prime};\widetilde{\sigma})}{p^{2}-p^{\prime
2}}\\\ \quad\times\left[e^{iT\left(p^{2}-p^{\prime
2}\right)/2m\hbar}-1\right]\widetilde{\psi}_{0}(p^{\prime})=\lambda\widetilde{\psi}_{0}(p)\,.$
(31)

We now rewrite Eq. (31) in a dimensionless form. Making the change of
variables

$\displaystyle u\equiv\sqrt{\frac{T}{4m\hbar}}\,p\qquad\text{and}\qquad
u^{\prime}\equiv\sqrt{\frac{T}{4m\hbar}}\,p^{\prime}\,,$ (32)

defining the dimensionless width $\varsigma$ of the precision function as

$\displaystyle\varsigma\equiv\sqrt{\frac{T}{m\hbar}}\,\widetilde{\sigma}\,,$
(33)

and introducing the dimensionless functions

$\displaystyle\mathcal{U}(u,u^{\prime};\varsigma)$
$\displaystyle\quad\equiv\sqrt{\frac{T}{4m\hbar}}\,U\left(\sqrt{\frac{4m\hbar}{T}}u,\sqrt{\frac{4m\hbar}{T}}u^{\prime};\sqrt{\frac{m\hbar}{T}}\varsigma\right)$
$\displaystyle\quad=\frac{u+u^{\prime}}{2}e^{-(u-u^{\prime})^{2}/\varsigma^{2}}\operatorname{erfc}\left(\frac{u+u^{\prime}}{\varsigma}\right)$
$\displaystyle\qquad\qquad-\frac{\varsigma e^{-2(u^{2}+u^{\prime
2})/\varsigma^{2}}}{2\sqrt{\pi}}$ (34)

and

$\displaystyle\varphi(u)\equiv\left(\frac{4m\hbar}{T}\right)^{1/4}e^{-iu^{2}}\widetilde{\psi}_{0}\left(\sqrt{\frac{4m\hbar}{T}}u\right)\,,$
(35)

we arrive at the following dimensionless integral eigenproblem:

$\int_{\mathbb{R}^{+}}du^{\prime}\,\frac{u+u^{\prime}-\mathcal{U}(u,u^{\prime};\varsigma)}{\pi(u^{\prime
2}-u^{2})}\sin(u^{2}-u^{\prime 2})\varphi(u^{\prime})\\\
=\lambda\varphi(u)\,.$ (36)

Figure 2: Behavior of the maximal eigenvalue $\Delta_{\text{max}}$ (solid blue
curve) of the eigenvalue equation (36) with respect to the (dimensionless)
width $\varsigma$ of the Gaussian precision function. The dash-dotted
horizontal red line represents the Bracken-Melloy bound (15). Inset: Logarithm
of $\Delta_{\text{max}}$ as a function of $\varsigma^{2}$ (solid blue curve).
The dashed green curve shows a Gaussian decay.

We then numerically solve the eigenvalue equation (36) and evaluate the
largest eigenvalue $\Delta_{\text{max}}$444The numerical method is based on
the approach originally presented in Ref. [1], and has been further discussed
in Section IIIB of Ref. [29] and the appendix of Ref. [12]., i.e. the maximal
backflow, for different values of the (dimensionless) width $\varsigma$. The
results are presented in Fig. 2. The solid blue curve shows the behavior of
$\Delta_{\text{max}}$ as a function of $\varsigma$, and the dash-dotted
horizontal red line represents the Bracken-Melloy bound, Eq. (15).

Our first observation is that $\Delta_{\max}$ approaches the Bracken-Melloy
bound as $\varsigma\to 0$. This means that for small enough widths, i.e., for
a Gaussian precision function that is sufficiently narrow in momentum space
(and, consequently, sufficiently broad in position space), the EF criterion
(10) allows for the same maximal backflow probability transfer as the standard
criterion (2). This fully agrees with the fact, explicitly noted in Ref. [33],
that in the limiting case of a precision function given by a Dirac
$\delta$-function in momentum space, the EF formulation (10) of QB reduces to
the standard one, Eq. (2), for positive-momentum states.

Our second observation is that $\Delta_{\text{max}}$ monotonically decreases
with $\varsigma$. This means that the EF criterion (10) is less efficient at
signalling the presence of QB at larger momentum widths of the precision
function. Furthermore, Fig. 2 strongly suggests that $\Delta_{\text{max}}$
vanishes in the limit $\varsigma\to\infty$.

At first sight, it might seem that the curve $\Delta_{\text{max}}(\varsigma)$
follows a Gaussian decay. This is however not the case, as is clear from the
inset in Fig. 2. The latter indeed shows that the logarithm of
$\Delta_{\text{max}}$ (solid blue curve) depends on $\varsigma^{2}$ in a more
intricate way than a simple linear dependence (dashed green curve).

Figure 2 allows us to estimate the parametric regime in which the EF criterion
(10) is capable of identifying QB. At the practical level, the backflow
probability transfer is only appreciable for $\varsigma\lesssim 1$. (We can
see from the data that
$\Delta_{\text{max}}=\frac{1}{2}\Delta_{\text{max}}^{(\text{BM})}$ for
$\varsigma\approx 1.29$.) This condition can, in view of Eq. (33), be
interpreted as follows: A measurement apparatus, characterized by a momentum
precision $\widetilde{\sigma}$, may be able to detect QB only if the
measurement is performed on a time scale $T$ satisfying

$\displaystyle T\lesssim\frac{m\hbar}{\widetilde{\sigma}^{2}}\,.$ (37)

##  5 Conclusion

We have investigated the compatibility of two formulations of quantum backflow
(QB) – on the one hand, the original formulation due to Bracken and Melloy
[1], with the backflow criterion given by Eq. (2), and, on the other hand, the
“experiment-friendly” (EF) formulation, with criterion (10), recently proposed
in Ref. [33]. In order to make a direct comparison, we applied both
formulations to the case of a free particle in a positive-momentum state. The
EF criterion, Eq. (10), involves a free parameter $\widetilde{\sigma}$,
playing the role of a momentum-space width of the precision function of a
measurement apparatus. Our main conclusion, in a nutshell, is that the two
formulations of QB are compatible only if $\widetilde{\sigma}$ is
substantially small, i.e., only if the measurement apparatus is sufficiently
precise in momentum space.

More specifically, we have considered a concrete example of a normalized
positive-momentum state, given by Eq. (16), that exhibits QB in the standard
sense of criterion (2). We have demonstrated the existence of a critical value
of the width $\widetilde{\sigma}$ above which the EF criterion (10) is not
satisfied, implying that, in the latter parameter range, the EF formulation of
QB is in conflict with the standard one.

Then, in the context of the EF formulation of QB, we have investigated the
maximal backflow probability transfer $\Delta_{\max}$ through a spatial point
over a fixed time interval. We have numerically determined
$\Delta_{\text{max}}$ as a function of the width $\widetilde{\sigma}$. Our
analysis indicates the following behavior of this function. As
$\widetilde{\sigma}$ tends to zero, $\Delta_{\max}$ approaches the Bracken-
Melloy bound, given by Eq. (15). Then, as $\widetilde{\sigma}$ increases, the
maximal backflow transfer decays monotonously and vanishes in the limit
$\widetilde{\sigma}\to\infty$. This allows us to identify the following
parametric regime in which the EF criterion (10) is capable of signaling the
occurrence of QB: The measurement apparatus has to operate on a sufficiently
short time scale $T$, satisfying condition (37), in order to make the
detection of QB practically feasible.

The EF formulation aims at providing a definition of QB that is better suited
to an experimental investigation as compared to the standard one. Therefore, a
brief discussion of the possibility of an experimental validation of the EF
backflow is in order. Checking the validity of Eq. (10a) experimentally
requires to measure the probability current $j_{t}(a)$ and, in addition, to
evaluate the momentum integral of $pf_{t}(a,p)$. As pointed out e.g. in Ref.
[1], the problem of measuring $j_{t}(a)$ can be mapped onto a more
conventional problem of measuring an electric current if one conducts the
experiment with electrically charged particles.555The relation between the
electric and probability currents is discussed in section 2.4 of Ref. [39].
Evaluating the momentum integral of $pf_{t}(a,p)$ is clearly a more
challenging problem, as it requires one’s ability to experimentally access the
phase-space distribution $f_{t}$, i.e. the Husimi function in our case. This
challenge however is not beyond the reach of cutting-edge experimental
techniques. As stated in Appendix A of Ref. [40], the Husimi function is the
expectation value of an observable and, as such, is a measurable quantity.
While more ingenious strategies may exist, an experimental measurement of the
wave function itself (using, e.g., techniques developed in Refs. [41, 42, 43,
44]) can in principle be used to reconstruct the Husimi distribution.

One of the main appeals of the EF formulation of QB, introduced in Ref. [33],
is that it offers a promising approach for studying QB in situations where a
particle moves in the presence of external forces and in systems of many
interacting particles. However, as we have shown in this paper, care must be
exercised when using the EF criterion (10): One must make sure that the
measurement apparatus is sufficiently precise in momentum space and operates
on short enough time scales.

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