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

Planck’s 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 beamsplitter 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 etal.

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.

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)
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 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.
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..