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It was previously demonstrated that a drop can bounce on a vibratin...
The author is the grandchild of Niels Bohr, the father of the Copen...
The double-slit experiment is the quintessential example of both in...
This is absolutely amazing, and cannot be underestimated. It shows ...
Watch this phenomenon from 0:16s in this video: [![](https://i.imgu...
They are also called Faraday waves, and have many cool applications...
De Broglie and later David Bohm suggested a different interpretatio...
To date, this analogy seems to break down because classical droplet...
This analogy between droplets on a fluid and quantum particles is i...
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FLUIDS
Quantum physics dropwise
Classical wave-driven particles can mimic basic quantum properties, but how far this parallel extends is yet to be
seen. Evidence for quantum-like mirages in a system of droplets moving on a fluid surface pushes the analogy into
many-body territory.
Tomas Bohr
I
nterference and superposition of particle
motion was, until recently, believed to
be unique to quantum mechanics. These
concepts are useful for describing extended
fields — or waves — whose effects can be
overlaid at each point in space, but seem
incompatible with localized particles
following well-defined orbits. This belief
was recently shown to be wrong for a class
of wave-driven particles via experiments on
millimetric silicon droplets bouncing on a
silicon bath
1
. Now, writing in Nature Physics,
Pedro Sáenz and colleagues
2
have shown that
a particle in the same system slowly builds
up a spatial probability distribution that is
closely correlated with the average wave field
it excites.
It may seem strange that a fluid
drop can actually ‘bounce’ on a fluid
surface without being swallowed by the
surrounding fluid. But, in fact, a thin
layer of air between the drop and the fluid
persists, constantly being renewed if the
oscillations are sufficiently fast
3
. When
driven violently enough, these drops will
start ‘walking’ across the surface
1
.
Curiously, this is closely connected with
a discovery made by Michael Faraday in
1831
4
. Forcing a dish containing a thin
plane fluid layer into vertical vibrations (for
example, using a violin bow), he noticed that
sufficiently strong vibrations would cause a
pattern of standing waves, which he called
crispations, to form on the fluid surface.
In other words, a fluid layer subjected to
vertical oscillations is intrinsically unstable.
If these oscillations are strong enough, the
fluid will spontaneously generate standing
waves, and just below this threshold the
surface will be extremely sensitive. So a
bouncing drop can create, if not a big splash,
then at least large and long-lived standing
surface waves — still without merging with
the fluid.
Droplets can even be propelled along the
fluid surface by these waves
1
. It is somewhat
counterintuitive that standing waves can
create motion. It is a bit like moving on
caterpillar tracks, sequentially laying down
a new segment in order to move forward. A
drop that happens, by chance, to bounce —
almost touching the surface — at a position
slightly displaced from where it took off
(emitting its last wave), will bounce on a
slightly tilted surface. This imparts a small
horizontal momentum to the drop. The
next bounce will thus create standing waves
centred at a displaced position, and, if the
decay time of these waves is sufficiently long,
this can lead to sustained horizontal motion.
The walking drop depends on its
standing wave for its motion and the wave
exists only because of the droplet. So the
particle and the wave form an inseparable
unit akin to the quantum description of
particles — in particular, the ‘pilot waves
introduced by de Broglie in 1924
5
just prior
to the discovery of quantum mechanics and
triggering its wave mechanical formulation.
How far can this analogy be taken?
As yet, we do not know. One of the first
striking observations with walking droplets
was spatial discretization. Placing the
vibrator and the walking drop on a rotating
table produces a system that closely
imitates a charged particle circulating
magnetic field lines
6
. Indeed, the walker’s
motion is changed from rectilinear to
circular, and, surprisingly, only certain
orbits are allowed — just like in the Bohr
model of the hydrogen atom. Replacing
Plancks constant by the wavelength of
the Faraday standing waves multiplied
by the mass and velocity of the drop, one
gets a sequence of radii matching the
Bohr–Sommerfeld quantization rules, the
so-called old quantum theory preceding
quantum mechanics proper. The full
quantum mechanical treatment gives
quantized orbits with similar mean radius,
but the details are different, because the
eigenstates do not correspond to well-
defined orbital radii.
To get closer to the heart of quantum
mechanics and challenge the statistical
Copenhagen interpretation, one can use
wave-driven particles to imitate individual
quantum processes in the hope of obtaining
the ‘realist’ model of quantum mechanics
that would have made Einstein and many
others so happy. Thus one should be able
to describe particles in a superposition of
eigenstates, like an entangled pair, or like
an electron or a photon passing through
the double-slit experiment. Indeed,
evidence for ‘quantum’ interference has
already been seen in a droplet version of
the double-slit experiment
7
, even though
one can easily observe through which slit
the droplet passes, as part of its wave field
can go though the other slit and create
interference (Fig.1). This, however, is not
correct:obviously walking droplets can be
influenced by their own wave field or that of
another droplet, but quantum interference is
something very special.
To determine the quantum probability
amplitude of going from one point of
measurement to another, all paths between
them have to be taken into account, each
contributing a probability amplitude
determined by the classical action for the
given path. In the drop experiments that is
Fig.1 | The walking droplet double-slit
experiment. The double-slit experiment became
emblematic of the interpretation of quantum
mechanics through the discussions between Bohr
and Einstein in 1927. A ‘walking droplet’ is seen on
its way across the surface of a shallow vibrating
layer of silicon oil. The triangular droplet emitter,
the barriers and the two rectangular slits can
be seen beneath the fluid surface. The walking
droplets closely resemble quantum particles
driven by a ‘pilot wave’, but how far the analogy
can be taken is presently unknown. Reproduced
from ref.
8
, APS.
NATURE PHYSICS | www.nature.com/naturephysics
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
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not the case: the path taken by the particle is
singled out and breaks the ‘path symmetry’.
For the double-slit experiment, the paths
through the two slits have a precise phase
difference depending on the difference in
length between the two paths — something
that obviously cannot be maintained
in experiment. This asymmetry can be
accentuated by putting in a separating wall
or a beamsplitter before the slits
8
, as is
typically done in optics experiments
9
. The
Schrödinger wave happily splits, but not so
for the walking drop.
Another way of imitating
basic quantum effects is to go beyond the
single particle to probe such intriguing
macroscopic quantum states as
Bose condensates and superconductivity.
In many-body systems, interactions
can give rise to complex states as
in the Kondo problem, where a
localized magnetic impurity radically
alters the low-temperature properties of an
electron gas.
Saenz etal.
2
have explored this relation,
extending earlier work on corrals
10
to show that a localized impurity can
strongly affect the superposition of basic
states governing the long-term motion
of a single particle. This can even lead
to ‘mirage’ effects, projecting from one
focal point of the elliptic corral to the
other. To go further and imitate features
of the spectacular macroscopic quantum
states, one needs to look more carefully at
systems with many wave-driven particles.
One promising result in this direction
is the observation of coherent states of
many droplets moving in long, narrow
channels
11
, sharing their wave fields and
moving at an elevated velocity. It would be
extremely interesting to know how closely
such systems can imitate their quantum
superconducting analogues.
Tomas Bohr
Department of Physics, Center for Fluid Dynamics,
Technical University of Denmark, Kongens Lyngby,
Denmark.
e-mail: tomas.bohr@fysik.dtu.dk
Published: xx xx xxxx
https://doi.org/10.1038/s41567-017-0015-6
References
1. Couder, Y., Protière, S., Fort, E. & Boudaoud, A. Nature 437,
208 (2005).
2. Sáenz, P. J., Cristea-Platon, T. & Bush, J. W. M. Nat. Phys. https://
doi.org/s41567-017-0003-x (2017).
3. Walker, J. Sci. Am. 238, 151–158 (June, 1978).
4. Faraday, M. Philos. Trans R.Soc. Lond. 121, 299–340 (1831).
5. de Broglie, L. Phil. Mag. 47, 446–458 (1924).
6. Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. Proc.
Natl Acad. Sci. USA 107, 17515–17520 (2010).
7. Couder, Y. & Fort, E. Phys. Rev. Lett. 97, 154101 (2006).
8. Andersen, A. et al. Phys. Rev. E 92, 013006 (2015).
9. Grangier, P., Roger, G. & Aspect, A. Europhys. Lett. 1, 173–179 (1986).
10. Harris, D. M., Moukhtar, J., Fort, E., Couder, Y. & Bush, J. W. M.
Phys. Rev. E 88, 011001 (2013).
11. Filoux, B., Hubert, M. & Vandewalle, N. Phys. Rev. E 92, 041004 (2015).
NATURE PHYSICS | www.nature.com/naturephysics
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

Discussion

Interesting fact: Tomas was the son of Aage Bohr and grandson of Niels Bohr. Both his father and his grandfather were Physics Nobel Laureates: - [Niels Bohr Nobel in Physics in 1922](https://www.nobelprize.org/nobel_prizes/physics/laureates/1922/bohr-facts.html) - [Aage Bohr Nobel in Physics in 1975](https://www.nobelprize.org/nobel_prizes/physics/laureates/1975/bohr-facts.html) This analogy between droplets on a fluid and quantum particles is intriguing, and it is clear that there are still unsolved problems here. In particular, (1) how would multiple droplets interact while moving on a fluid - will they resemble other quantum effects? (2) what about spin? (3) and lastly, can this classical analog mimic the collapse of the wave function under any circumstances? Many experts in the field believe that the answer for (3) is a sound "no", and this is why it is only an analogy, and nothing more.. It was previously demonstrated that a drop can bounce on a vibrating fluid bath and exhibit many quantum phenomena (despite being a classical object). This video is a good demo of these phenomena: [![](https://i.imgur.com/GcYBv5d.png)](https://www.youtube.com/watch?v=nmC0ygr08tE) This is absolutely amazing, and cannot be underestimated. It shows that even though fluid dynamics is a continuous theory, its nonlinearity can reproduce discrete parameters in special cases. This makes one wonder if the same thing could be true in a more fundamental theory of physics, where nonlinearity will be responsible for many complex quantum phenomena that we measure... The double-slit experiment is the quintessential example of both interference and quantum superposition. Learn more here: [Double-slit Experiment.](https://en.wikipedia.org/wiki/Double-slit_experiment) They are also called Faraday waves, and have many cool applications in Nature. For example, alligators use the same phenomenon to call their mates. You can read more about it here: [Faraday wave.](https://www.wikiwand.com/en/Faraday_wave) To date, this analogy seems to break down because classical droplet don't exhibit any effect analogous to the wave function collapse. Watch this phenomenon from 0:16s in this video: [![](https://i.imgur.com/XT3IrJp.png)](https://youtu.be/nmC0ygr08tE?t=16s) De Broglie and later David Bohm suggested a different interpretation to quantum mechanics, contrary to the conventional "Copenhagen interpretation". In their interpretation, there is a quantum wave that is constantly interacting with the particle. As the particle moves, it affects the quantum wave, and mutually the quantum wave guides the particle in its motion. This interpretation is capable of explaining many quantum phenomena in a more intuitively manner, in particular the double-slit experiment. In the double-slit experiment the wave patterns are created precisely because the wave is guiding the particle. However, this interpretation never became popular due to various reasons. The interpretation is called the De Broglie-Bohm theory, Bohmian mechanics, or Pilot Wave theory. The author is the grandchild of Niels Bohr, the father of the Copenhagen interpretation of quantum mechanics and one of the leading contributors to the theory.