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# Experiment-friendly formulation of quantum backflow

Marek Miller  School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore  Centre of New Technologies, University of
Warsaw, Poland<EMAIL_ADDRESS>Woo Chee Yuan<EMAIL_ADDRESS>School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore  Rainer Dumke  School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore  Centre for Quantum
Technologies, National University of Singapore, Singapore  Tomasz Paterek
School of Physical and Mathematical Sciences, Nanyang Technological
University, Singapore  MajuLab, International Joint Research Unit UMI 3654,
CNRS, Universite Cote d’Azur, Sorbonne Universite, National University of
Singapore, Nanyang Technological University, Singapore  Institute of
Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and
Informatics, University of Gdańsk, Poland

###### Abstract

Quantum backflow is usually understood as a quantum interference phenomenon
where probability current of a quantum particle points in the opposite
direction to particle’s momentum. Here, we quantify the amount of quantum
backflow for arbitrary momentum distributions, paving the way towards its
experimental verification. We give examples of backflow in gravitational and
harmonic potential, and discuss experimental procedures required for the
demonstration using atomic gravimeters. Such an experiment would show that the
probability of finding a free falling particle above initial level could grow
for suitably prepared quantum state with most momentum downwards.

##  1 Introduction

A wave function of a quantum particle has physically observable
characteristics that can be local or global. The probability of finding the
particle in a specific region of space or the probability current are examples
of local characteristics, which can be determined given access to only small
part of the wave function. In contradistinction, the momentum is a property of
the entire wave function, e.g. requires the determination of the de Broglie
wavelength. Already at this level of generality, it is clear that the
probability current and the momentum of a quantum particle may behave very
differently.

Quantum backflow (QB) is an interference effect that arises from this
observation. In order to understand the statement better, and to simplify the
subsequent analysis, let us suppose that a particle in the one-dimensional
space interacts with a potential that depends only on the particle’s position.
Intuitively, we may think that if the momentum distribution concentrates
within the positive half-line, the probability current, too, will be non-
negative. It turns out that this is not the case for a suitably chosen quantum
state. In a sense, the probability flows ‘backwards’. Hence the term “quantum
backflow”.

QB was first studied by Allcock in his work on the arrival time in quantum
mechanics [2] (see also Kijowski [17] for an early discussion of QB). Later,
Bracken and Melloy [7] gave a detailed analysis of QB as an eigenvalue
problem. The analysis was rigorously phrased in the mathematical language of
integral operators on separable Hilbert space in [24]. In the same paper, a
numerical approximation of the optimal QB state was given. See also [16, 28]
for intresting case studies. QB in systems interacting with linear potential
was studied in [21]. Recently, there have been attempts at analysing QB in the
relativistic setting [20, 25, 3], in the setting of quantum particle decay [9,
13], as well as the attempts at describing quantum backflow in dissipative
[22] and many-particle systems [4]. The phenomenon was extended into phase
space in Ref. [14] and systems suitable for observing QB in an experiment were
discussed in Refs. [27, 15].

Among others, Palmero _et al_. [23] proposed an experimental scheme that
“could lead to the observation of quantum backflow” in
$\phantom{}{}^{7}\text{Li}$ Bose-Einstein condensate. To the best of our
knowledge, however, QB has not yet been confirmed experimentally. Note that in
a recent experimental work by Eliezer  _et al._ [10], “optical backflow” in
transverse momentum of a beam of light was reported. The present study,
however, focuses on quantum backflow of individual non-relativistic quantum
particles. We extend the customary definition of QB to states with non-zero
probability of measuring negative momentum. Hence, our approach should be
applicable to realistic experimental situations. We study systems exhibiting
QB in gravitational field near the surface of the Earth, as well as
interacting with harmonic potential. We also comment on possible experimental
verification of QB using atomic gravimeters.

##  2 Quantum backflow

Our definition of quantum backflow conveys how necessary conditions for the
probability current that follow from the classical equations of motion are no
longer satisfied in the quantum regime.

Let us focus on the system of a lone particle in the one-dimensional space. We
set the vertical direction with the $x$ axis pointing downwards. Suppose the
particle interacts with arbitrary potential $V(x)$. We examine its dynamics
from classical and quantum points of view with initial conditions as similar
as possible. The quantum system at time $t$ is fully described by its wave
function $\psi_{t}(x)$. The classical model requires simultaneous knowledge of
position and momentum, whose precise estimation is famously forbidden by the
uncertainty principle. We therefore propose an operational approach in which
distribution of position and momentum is estimated with finite precision, and
given by the probability density function

$f_{t}(x,p)=\left|\langle\phi|W(x,p)|\psi\rangle\right|^{2},$ (1)

where $W(x,p)$ is the Wigner-Moyal transform of $\psi$ and $\phi$ (see e.g.
Eq. (6.63) in [8]):

$\langle\phi|W(x,p)|\psi\rangle=\\\
=\frac{1}{2\pi\hbar}\int\limits_{-\infty}^{\infty}e^{-\frac{i}{\hbar}py}\phi^{*}(y-\frac{x}{2})\,\psi(y+\frac{x}{2})\,dy.$
(2)

The function $\phi$ represents the finite precision of the measurement
apparatus and, for example, can be a Gaussian distribution, centred at zero,
with finite width $\sigma_{\phi}$. The marginals of $f_{t}(x,p)$ agree with
densities of position and momentum, derived from $\psi_{t}(x)$, "up to
$\sigma_{\phi}^{2}$ ", i.e.:

$\displaystyle P_{t}(x=x_{0})$
$\displaystyle=\int\limits_{-\infty}^{\infty}f_{t}(x_{0},p)dp=(|\psi|^{2}*|\phi|^{2})(x_{0}),$
(3) $\displaystyle P_{t}(p=p_{0})$
$\displaystyle=\int\limits_{-\infty}^{\infty}f_{t}(x,p_{0})dx=(|\tilde{\psi}|^{2}*|\tilde{\phi}|^{2})(p_{0}),$
(4)

where $\tilde{\psi}(p)$ and $\tilde{\phi}(p)$ are Fourier transforms of $\psi$
and $\phi$, respectively, and the symbol ’$*$’ stands for convolution.

In other words, the family of operators
$W(x,p)^{\dagger}|\phi\rangle\langle\phi|W(x,p)$, defined on the phase space
$(x,p)\in\mathbb{R}^{2}$, is a positive operator-valued measure that allows
experimental estimation of the joint probability distribution $f_{t}(x,p)$ of
position and momentum in the state $\psi$, with a finite precision given by a
square-integrable function $\phi$.

Let us now derive probability currents in the quantum and classical models
corresponding to the rate of change of probability of finding the particle
above the level $x=a$. In the quantum case, this is a textbook exercise
leading to the familiar formula

$j_{t}(a)=-\frac{d}{dt}P_{t}(x\leq a)=\\\
=\frac{\hbar}{m}\mathfrak{Im}\left(\psi_{t}^{*}(a)\,\frac{d}{dx}\psi_{t}(x)\Bigr{|}_{x=a}\right),$
(5)

where $\mathfrak{Im}$ stands for the imaginary part of a complex number.

Let us fix $t$ for now and consider a putative classical system, with the
distribution of position and momentum given by $f_{t}(x,p)$. The distribution
evolves for a short time $\tau$, $t\leq\tau\leq t+\Delta t$ according to the
Hamilton equations of motion:
$\dot{x}(\tau)=p/m,\quad\dot{p}(\tau)=-\frac{dV}{dx}$. This implies that,
unlike in the quantum case, the probability current of the classical system,

$(j_{\textrm{cl}})_{t}(a)=-\frac{d}{d\tau}P_{\textrm{cl}}\left(x(\tau)\leq
a\right)\Big{|}_{\tau=t},$ (6)

must be bounded from below:

$(j_{\textrm{cl}})_{t}(a)\geq\frac{1}{m}\int\limits_{-\infty}^{0}p\,f_{t}(a,p)\,dp$
(7)

(see Appendix for the detailed proof). We can say that the probability of
finding the classical particle above the line $x=a$ cannot grow faster than a
certain quantity derived from the distribution of only negative momenta. In
particular, if $f_{t}(x,p)=0$ for $p<0$, we get that
$(j_{\textrm{cl}})_{t}(a)\geq 0$, i.e. the direction of the momentum and the
probability current coincide. In this particular case, it leads to the usual
definition of QB given by the following statements about a wave function of a
quantum system in one-dimensional space [6]: a) $\tilde{\psi}_{t}$ contains
only positive momenta; b) there exists $a\in\mathbb{R}$, for which
$j_{t}(a)<0$. To facilitate the analysis of QB for arbitrary states, we say
that QB is a situation where the inequality (7) no longer holds.

######  Definition 1.

The quantum backflow takes place at point $x=a$ and at time $t$, if

$j_{t}(a)<\frac{1}{m}\int\limits_{-\infty}^{0}p\,f_{t}(a,p)\,dp,$ (8)

where the probability current $j_{t}(a)$ is given by Eq. (5) and the function
$f_{t}(x,p)$ by Eq. (1).

Note that by taking a formal limit of the precision function converging to the
Dirac delta distribution in the momentum representation,
$\tilde{\phi}(p)\rightarrow\delta(p)$, we can see from Eq. (2) that
$f_{t}(x,p)\rightarrow|\tilde{\psi}(p)|^{2}$, at any point $x$. Hence, in this
limiting case, and for a wave function such that $\tilde{\psi}(p)=0$ for
$p<0$, the inequality (8) reduces to the standard definition of QB, requiring
only $j_{t}(a)<0$.

It is also worth mentioning that Eq. (8) is a condition independent of whether
the system undergoes quantum scattering. For example, it is easy to see that
an initial superposition of plain waves,

$\psi(x)=Ae^{ikx}+Be^{-ikx},\quad x<0,$ (9)

scattered back on a potential step, does not exhibit quantum backflow. Unlike
scattering, quantum backflow is an instantaneous property of the particle’s
quantum state, namely its probability current exceeding a classical bound,
regardless of the underlying dynamics.

Figure 1: Quantum backflow in gravity. The initial state is a superposition of
two Gaussian wave functions with amplitudes given in (12), describing Rubidium
atom. Left panel: The dashed line gives the lower bound on the classical
probability current. The solid line is the quantum probability current. QB
takes place in the interval when the solid line is below the dashed line, see
Eq. (8). Right panel: As a consequence, the probability $P(x<0)$ of finding
the particle above the initial level of $x=0$ increases despite small
contributions from negative momenta. Solid line gives the probability in the
presence of the gravitational field. For comparison, the dashed line
represents the free particle.

##  3 Examples

We give two concrete examples of QB according to Definition 1. Both involve
superposition of Gaussian states and therefore naturally contain contributions
from negative momenta. The first example is a particle in a linear potential
as an approximation of the Newtonian gravity close to the Earth’s surface. The
second example considers quadratic (harmonic) potential.

###  3.1 Gravitational potential

Suppose the particle interacts with the potential $V(x)=-mgx$, for $g\geq 0$.
Recall that by our convention, the direction of the $x$ axis and the direction
of the gravitational force coincide. We choose the initial level $x=0$ above
the surface. Of course, by putting $g=0$, we also cover the case of a free
particle on the real line.

Consider the initial wave function, at time $t=0$, being a superposition of
two Gaussian states centred at $x=0$, with the same spread $\sigma$ but
different mean momenta. If the corresponding quantum particle is free, the
wave function at time $t$ reads:

$\psi_{\mathrm{free}}(x,t)=\sum_{n=1}^{2}\frac{B_{n}}{\sqrt{4\sigma^{2}+2i\frac{h}{m}t}}\times\\\
\times\exp\bigg{(}\frac{i}{\hbar}p_{n}\big{(}x-\frac{p_{n}}{2m}t\big{)}-\frac{(x-\frac{p_{n}}{m}t)^{2}}{4\sigma^{2}+2i\frac{h}{m}t}\bigg{)},$
(10)

where $p_{1},p_{2}$ are the mean values of momentum for each branch of the
superposition and $B_{1},B_{2}$ are the corresponding probability amplitudes.
In the presence of the linear potential $V(x)$, the wave function accelerates
as a result of the gravitational force, and at time $t$ it takes the form
[26]:

$\psi(x,t)=e^{-\frac{i}{\hbar}(-mgtx+\frac{1}{6}mg^{2}t^{3})}\,\psi_{\mathrm{free}}(x-\frac{1}{2}gt^{2},t).$
(11)

Let us now choose realistic values of the parameters (see next section for
further discussion of a possible experiment). E.g., let us take the Rubidium
atom of mass $m\approx 1.4\times 10^{-25}$ kg, described by the wave function
with $\sigma=1\mu$m. The mean momenta are set to $p_{1}=0$ and $p_{2}=2\hbar
k$, where the wave number $k=2\pi/\lambda$ with $\lambda=780$ nm. The
following amplitudes

$B_{1}\approx 1.18\times 10^{-3},\ B_{2}\approx 4.42\times 10^{-4},$ (12)

numerically optimise the effect of QB. The precision function is set to be a
Gaussian with standard deviation $\sigma_{\phi}=0.1\mu$m.

In Fig. 1, we show the probability current and the overall probability of
finding the particle above the initial line $x=0$ as functions of time. We set
$g=9.8\,\text{m}\cdot\text{s}^{-2}$. It is clear that for $t$ such that
approximately: $23.7\mu$s $\leq t\leq 39\mu$s, the probability current
satisfies the inequality (8), i.e. QB takes place. During that time, the
numerical value of the integral $\int_{-\infty}^{0}|\tilde{\psi}(p)|^{2}\,dp$,
i.e. the contribution of “negative momenta” to the backflow state, ranges from
$0.23$ to $0.13$. Here, our approach allows us to separate the contribution of
the negative momenta from the genuine quantum backflow. Despite the particle’s
free fall, the quantum probability of finding the particle above the initial
level unmistakably _increases_.

###  3.2 Harmonic potential

Our second example is a particle in quadratic potential, due to its wide
applicability. We again consider Rb atom, but this time in a harmonic trap
with frequency $\nu=10$ kHz [5]. It is generally known that coherent states of
a quantum harmonic oscillator take the form of a Gaussian packet in position
representation [18]. Hence, we take as an initial state the superposition of
two coherent states $|\psi\rangle=a|\alpha\rangle+a|\beta\rangle$. Numerical
optimisation of QB leads to the following parameters: $a\approx 0.73$,
$\alpha=e^{i(0.9\pi-\omega t)}$, $\beta=9e^{i(0.55\pi-\omega t)}$, where
$\omega=2\pi\nu$. Fig. 2 shows the corresponding classical bound on the
probability current and the quantum prediction for QB.

Figure 2:  Quantum backflow in a harmonic trap. Rb atom in the initial
superposition of coherent states with parameters given in the main text gives
rise to the quantum probability current (solid line) which is below the
classical lower bound (dashed line). This signifies QB for approximately
$(230,239)\mu$s. Probability current was evaluated at
$x=\sqrt{\frac{2\hbar}{m\omega}}\cos{(0.55\pi)}$.

##  4 Experimental proposal

The approach proposed here is based on the measurement of probability current
and generalised joint measurement (with finite precision) of position and
momentum. The latter determines the right-hand side of Eq. (8) and the former
its left-hand side. Probability current is a convenient theoretical quantity
in this problem because in the derivation of (8) Hamilton’s equations reduce
the classical bound to the integral over momentum only. In more practical
terms, the measurement of the probability current requires estimation of
particle’s position at two nearby times and division by the corresponding time
interval. This can be simplified to only one estimation of particle’s position
by using the bound of Eq. (A) in the Appendix.

The essence of the present proposal is to establish the initial joint
probability distribution of position and momentum that is used to compute the
classical bound. Since this might be experimentally quite demanding, we offer
one more route to the verification of QB, based on independent measurements of
initial position and momentum distributions. Such measurements do not give any
information about possible position-momentum correlations. Accordingly, we
propose to consider all joint distributions $f_{t}(x,p)$ compatible with these
marginals and take the worst case upper bound in Eq. (A).

Finally, a concrete setup capable of measuring QB could be constructed using
atomic gravimeters. High precision estimation of position distribution has
been realised with electron [12] as well as optical microscopy [1, 19] of the
atomic cloud; and momentum distribution can be established from the time-of-
flight measurements [11]. Depending on the way the atoms are trapped these
setups offer initial width of the wave function in the range from micrometers
to millimetres. We have verified numerically that QB is present for all these
spreads with the order of magnitude of the bound and the quantum probability
current scaling like inverse square of the width.

##  5 Conclusions

We presented an analysis of QB for states with admissible negative momentum.
Unlike the standard definition of QB, Def. 1 is applicable not only to wave
functions with positive support in momentum representation. Broadly speaking,
our idea is to compare the classical evolution of the joint probability
distribution of position and momentum (known with finite precision) with the
quantum dynamics. The classical system would evolve for an infinitesimally
short time according to Hamilton’s equations of motion, starting with the same
initial conditions. We defined QB as a situation when the probability current
exceeds what might be possible for its classical counterpart.

In particular, we showed that a relatively easy to prepare superposition of
two Gaussian packets exhibits QB during free-fall in the uniform gravitational
field near the Earth’s surface. Of course, the Ehrenfest theorem guarantees
that the average position of the quantum particle follows the classical
trajectory. Nevertheless, the probability of locating the particle above the
initial level displays ‘antigravitational’ quantum backflow.

##  6 Acknowledgments

We would like to warmly thank Referees for their insightful comments and
questions.

This work is supported by the Polish National Agency for Academic Exchange
NAWA Project No. PPN/PPO/2018/1/00007/U/00001.

##  Appendix A Appendix

Here, we give a proof of the inequality (7).

Once again, we consider the distribution of position and momentum given by
$f_{t}(x,p)$ that evolves classically for a brief time $\tau$, $t\leq\tau\leq
t+\Delta t$. If our system were initially at point $x$, then
$x(\tau)=x+\frac{p}{m}(\tau-t)$. The probability $P(x(\tau)\leq a)$ of finding
the particle in the region “above” the line $x=a$ would be

$\displaystyle P(x(\tau)\leq a)$ $\displaystyle=$ $\displaystyle P(x\leq
a-\frac{p}{m}(\tau-t))$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{a-\frac{p}{m}(\tau-t)}\int\limits_{-\infty}^{\infty}f_{t}(x,p)\,dp\,dx$
$\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{a}\int\limits_{-\infty}^{\infty}f_{t}(x-\frac{p}{m}(\tau-t),p)\,dp\,dx.$

Now, the probability current of the classical system,

$(j_{\textrm{cl}})_{t}(a)=-\frac{d}{d\tau}P\left(x(\tau)\leq
a\right)\Big{|}_{\tau=t},$ (14)

can be expressed in terms of the probability density function $f_{t}(x,p)$:

$\displaystyle(j_{\textrm{cl}})_{t}(a)$ $\displaystyle=$
$\displaystyle-\frac{d}{d\tau}\int\limits_{-\infty}^{a}\int\limits_{-\infty}^{\infty}f_{t}(x-\frac{p}{m}(\tau-t),p)\,dp\,dx\Big{|}_{\tau=t}$
(15) $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{a}\int\limits_{-\infty}^{\infty}\frac{p}{m}\frac{\partial}{\partial
x}\,f_{t}(x,p)\,dp\,dx$ $\displaystyle=$
$\displaystyle\frac{1}{m}\int\limits_{-\infty}^{\infty}pf_{t}(a,p)\,dp,$

where the last equality is a consequence of the fact that
$\lim_{x\rightarrow\pm\infty}f_{t}(x,p)=0$. This immediately yields the
inequality:

$(j_{\textrm{cl}})_{t}(a)\geq\frac{1}{m}\int\limits_{-\infty}^{0}pf_{t}(a,p)\,dp.$
(16)

From a practical point of view, in order to save on measurements, it would be
more desirable to derive the classical upper bound for the probability of
finding the particle above the initial level instead of the lower bound on the
probability current. For this we use Eq. (A) and note the following inequality

$\displaystyle P(x(\tau)\leq a)$
$\displaystyle\leq\int\limits_{-\infty}^{a-\frac{p}{m}(\tau-t)}\int\limits_{-\infty}^{0}f_{t}(x,p)\,dp\,dx$
$\displaystyle+\int\limits_{-\infty}^{a}\int\limits_{0}^{\infty}f_{t}(x,p)\,dp\,dx,$
(17)

which is simply a consequence of non-negativity of $f_{t}(x,p)$. Hence,

$\displaystyle P(x(\tau)\leq a)$ $\displaystyle\leq P(x\leq a)$
$\displaystyle+\int_{a}^{a-\frac{p}{m}(\tau-t)}\int\limits_{-\infty}^{0}f_{t}(x,p)\,dp\,dx.$
(18)

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