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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.
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