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FLUID S

Quantum phy sics dr op wise

Classical w ave-driv en particles can mimic basic quantum properties, but how far this par allel extends is y et to be

seen. Evidenc e for quantum-lik e mirag es in a system of dr oplets moving on a ﬂuid surfac e pushes the analogy into

many-body t erritory .

T omas Bohr

I

nterf erence and su perp osition o f particle

motion was, un til recently , believed to

be unique to quan tum mecha nics. These

concep ts are useful for describing ext ended

fields — or wa ves — whose effec ts can be

over laid at each poin t in space, but seem

incom patible with localized particles

follo wing well-defined orbits. This belief

was recen tly shown to be wro ng for a clas s

of wa ve-driven particles via experiments on

millimetric silicon droplets boun cing on a

silicon bath

1

. No w , wr iting in N a tur e Phys ics ,

P edro Sáenz and colleagues

2

hav e shown that

a particle in the same system slo wly builds

up a spa tial probabili ty distribution tha t is

closely corr elated with the a verage wav e field

it exci tes.

I t may seem strang e that a fluid

drop ca n actually ‘bounce ’ on a fluid

surface withou t being swallowed by the

surroun ding fluid. But, in fact, a thin

layer o f air between the drop an d the f luid

persists, cons tantly being ren ewed if the

oscillations ar e sufficiently fast

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. When

driven violen tly enough, these drops will

start ‘walking’ acr oss the surface

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.

Curious ly , this is c losely connected with

a discovery made by Michae l Farada y in

1831

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. Fo rcing a dish co ntaining a thin

plane fluid la yer into v ert ical vibra tions (for

exam ple, using a violin bow), he n oticed that

sufficiently stro ng vibra tions would ca use a

patt ern of standing wa ves, which he called

crispation s, to form on the fluid surface.

In o ther words, a fluid la yer subjected to

vertical osci llation s is intrinsically unstab le.

If these oscillations a re stron g enough, the

fluid will spontaneous ly genera te standing

waves, a nd just belo w this threshold the

surface will be ext reme ly sensitive. So a

bouncing dr op can crea te, if not a b ig splash,

then at least lar ge and lon g-lived standing

surface waves — s till w itho ut merging with

the fluid.

Drop lets can even be propelled along the

fluid surface by these wav es

1

. I t is somewhat

coun terintui tive that stan ding waves can

creat e motion. I t is a bi t like moving o n

caterpillar tracks, sequen tially laying down

a new segment in or der to move f orward. A

drop tha t hap pens, by chan ce, to bounce —

almost touc hing the surface — at a position

slightly disp laced from where it took o ff

(emitting i ts last wav e), will b ounce on a

slightly tilted surface. Thi s imparts a small

horizon tal momen tum to the drop . The

next bounce will thus crea te standing wa ves

centr ed at a displaced position, an d, if the

decay time of these wa ves is sufficiently long,

this can lead to sustained ho rizontal motio n.

The walking drop depends o n its

standing wa ve for i ts motion a nd the wave

exists only becau se of the droplet. So the

particle and the wav e form an insepara ble

unit akin to the quan tum descriptio n of

particles — in particu lar , the ‘p ilot waves ’

intr oduced by de Broglie in 1924

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just p rior

to the discovery of quan tum mecha nics and

triggering its wav e mechanical form ulation.

Ho w far can this analogy be taken?

As yet, w e do not know . One of the first

striking obser vation s with walking droplets

was spatial discretiza tion. Placing the

vibra tor and the walking dr op on a ro tating

table pr oduces a system that c losely

imitat es a charged particle cir cul atin g

magnetic field lines

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. Indeed, the walker’ s

motion is c hanged fro m rectilinear to

circular , and, surprisingly , only certain

orbi ts are allow ed — just like in the Bohr

model of the h ydrogen a tom. Replacing

Planck ’ s co nstan t by the wav elength of

the Fa raday standing wa ves mul tiplied

by the mass a nd velocity of the dr op , one

gets a sequence of radii ma tching the

Bohr–Sommerfeld q uantiza tion rules, the

so-called old quantum theo r y preceding

quan tum mechanics p roper . The fu ll

quan tum mechanical trea tment gives

quan tized orbits wi th similar mean radius,

but the details a re differen t, because the

eigensta tes do not corr espond to well-

defined orbital radii.

T o get closer to the heart of q uant um

mechanics an d challenge the statistical

‘ Copenhagen interpr etation ’ , one can use

wave-dri ven particles to imitate in dividual

quan tum pr ocesses in t he hope o f obtaining

the ‘ realist’ m odel of quan tum mechanics

that wo uld have made Ein stein and man y

others so ha ppy . Thus o ne should be able

to describe particles in a superp osition o f

eigensta tes, like an en tangled pair , o r like

an electron o r a photo n passing through

the double-sli t experiment. Indeed,

evidence for ‘ quan tum ’ interfer ence has

already been seen in a droplet ver sion of

the double-sli t experiment

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, even though

one can easily observe through which slit

the drop let passes, as part of its wav e field

can go though the other s lit and crea te

interf erence (Fig. 1 ). This, how ever , is no t

correct:ob viously walking dro plets can be

influenced by their own wa ve field or tha t of

another dr oplet, bu t quan tum interfer ence is

something very specia l.

T o determine the quan tum pr obability

am plitude o f going fro m one poin t of

measuremen t to ano ther , all paths between

them hav e to be ta ken in to accoun t, each

con tributing a pr obability am plit ude

determined by the classical action fo r the

given path. I n the drop experiments tha t is

Fig.1 | T he walking droplet double-slit

experiment. The double-slit experiment became

emblematic of the interpretation of quantum

mechanics through the discussions betw een Bohr

and Einstein in 192 7 . A ‘walking droplet’ is seen on

its way acr oss the surface of a shallow vibr ating

layer of silic on oil. The triangular droplet emitter ,

the barriers and the two r ectangular slits can

be seen beneath the fluid surface. T he walking

droplets closely r esemble quantum particles

driven by a ‘pilot w ave’, but how f ar the analogy

can be taken is pr esently unknown. Repr oduced

from r ef.

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, APS.

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