The Casimir Effect - University of St Andrews

The
Casimir
Effect
Exploring
and
engineering
the
quantum
vacuum
Chun Xiong, Thomas Philbin, Tom Kelsey,
Ulf Leonhardt and Steve Linton

Casimir
forces
Two mirrors
Casimir, 1948:
(Also van der Waals,
London, Polder, …)
Lamoreaux, 1997(!). First unambiguous
demonstration of the Casimir force (sphere and plate).
Distances ~ 100 nm, Forces ~ 100 pN.
Capassso group, Harvard

Casimir
force
for
realis:c
materials
–
Lifshitz
theory
Two mirrors
Casimir, 1948:
For a~10nm, f ~ 1atmosphere.
Lifschitz, 1955:
A EE
= r −1 E1
r −1 E 2
e 2aw −1,
A BB
= r −1 B1
r −1 B 2
e 2aw −1,
€
w = u 2 + υ 2 + κ 2 ,
€
r E1
= µ 1(icκ)w − w 1
µ 1
(icκ)w + w 1
,
€
w 1
= u 2 + υ 2 + ε 1
(icκ)µ 1
(icκ)κ 2 .
€
r B1
= − ε 1 (icκ)w − w 1
ε 1
(icκ)w + w 1
,
€
€

Casimir
force
for
realis:c
materials
–
Lifshitz
theory
ω
is
a
frequency
associated
with
the
oscilla:on
of
a
classical
dipole
ε(ω)
is
the
electric
permiDvity
of
the
material
μ(ω)
is
the
magne:c
permeability
of
the
material
u,
v
and
κ
are
also
frequencies
–
in
prac:ce
we
integrate
over
(what
we
guess
are)
realis:c
finite
ranges
ω 2 
=
u 2 
+
v 2 
+
κ 2 ,
so
ω
describes
a
radius
in
(u,v,κ)‐space
Calcula:ons
are
(rela:vely)
straighNorward
when
we
have
uniform
media,
i.e.
ε
and
μ
are
constant

Casimir
forces
and
engineering
Casimir forces are already being seen in
MEMS (micro-electro-mechanicalsystems),
causing malfunctions –
‘stiction’.
(Nature News Feature, 2007)
Casimir forces will become even
more significant at smaller scales.
How large are the Casimir forces?
How can they be manipulated?

Manipula:ng
the
Casimir
force
Dielectric sandwich
Dzyaloshinskii, Lifshitz and Pitaevskii Adv. Phys. 10 (1961) 165
Repulsive force if ε 1
< ε 2
< ε 3
in the relevant (imaginary) frequency range.
€
Munday, Capasso and Parsegian, Nature 457 (2009) 170

Manipula:ng
the
Casimir
force
Magnetic mirror
Two mirrors
Conventional/magnetic mirror
f = cπ 2
240a 4
f = − 7cπ 2
1920a 4
Boyer, 1974
€
€
Metamaterials
Magnetic mirror now built,
for narrow bandwidth
Schwanecke et al. J. Op. A 9 (2007) L1

Quantum
Levita:on
Suppose one mirror in the Casimir cavity is a 500nm thick aluminium foil (much thicker
than the optical skin depth).
With a repulsive Casimir force, an effective separation a’ of around 500nm would
balance the gravitational force on the foil.
Force:
The foil would levitate
on vacuum fluctuations.
Philbin & Leonhardt 2007

Challenge:
intui:on
versus
Lifshitz
theory

Moving
media
Simplest case
(Also simplest model
of a nanomachine)
?
What are the Casimir forces?
How is the perpendicular
force modified by the motion?
What is the lateral force –
“Quantum Friction”?
Thirty years of contradictory claims on this problem.
Agreement (almost) that there is quantum friction unless the plates are perfect mirrors.
Barton (1996) found zero lateral force on the plates in a very simplified model.
Authors disagree on the magnitude of the Casimir forces; most try approximations.
Answer is a sum of contributions from modes with E– and B–polarizations. (Wrong.)

Lateral
Casimir
force?
The moving plate is like a non-moving anisotropic plate.
It is equivalent to a particular non-moving bi-anisotropic plate.
Suppose we build such a bi-anisotropic plate. Then can
there be a unidirectional lateral Casimir force on the plates?
This would seem to allow the extraction
of unlimited energy from the quantum vacuum.
Lateral
Casimir
force
–
Casimir
torque
Two non-aligned birefringent crystals experience a torque
that tries to align them.
[Barash, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 12 (1978) 1637]
[Philbin and Leonhardt, Phys. Rev. A 78 (2008) 042107]

The
Casimir
forces
[Philbin and Leonhardt, New J. Phys. 11, (2009) 033035]
Stress tensor is diagonal: no quantum friction.
xx–component of stress tensor gives perpendicular force per unit area on plates:
A XY
= r −1 X1
r −1 Y 2
e 2aw −1, X ,Y ∈ { E,B}.
f = c
4π 3
∞
∫
dκ
€
0
∞
∫
−∞
∞ ⎡
du ∫ dυ w A EE
+ A
⎢
BB
⎢
−∞
⎣ A EE
A BB
( ) 2 − A EB
+ A BE
( u 2 + υ 2 − iκuβ) 2 − A EB
A BE
w 2 υ 2 β 2
( ) u 2 + υ 2 − iκuβ
( )w 2 υ 2 β 2 ⎤
⎥
⎥ .
⎦
Static result (1955):
β = 0 :
f = c
4π 3
∞
∫
0
dκ
∞
∫
−∞
du
∞
∫
−∞
dυ w ( A −1 −1
EE
+ A BB ),
For dielectrics, motion adds an attractive component to the force compared to static case.
With a magnetic response this component due to the motion can be attractive or repulsive.
€

Casimir
forces
for
spheres
and
cylinders
Boyer, 1968. Casimir stress on a perfectly conducting
sphere is infinite.
Further regularization: force is repulsive!
No experiments. Theorists not agreed on what’s going on.
Casimir stress on a perfectly conducting cylinder
also diverges.
Further regularization: force attractive.

Casimir
forces
for
non‐homogeneous
media
We chose to consider a new situation:
QV
ε 1 is homogeneous as before
QV is the quantum vacuum
x
ε 2 varies with distance from the ε 1 boundary, becoming
constant at the QV boundary
Initially we considered exponential decay:
ε 2 (x) = ae -bx
Again, no experiments -- theorists not agreed on what’s going on.
We are at, or beyond, the limits of Lifshitz theory.
We are certainly beyond the limits of Casimir theory.

Casimir
forces
for
non‐homogeneous
media
Using Lifshitz theory we
found that the key
component of the stress
tensor gave an infinite value
when integrated.
Either we had made a
mistake, or Lifshitz theory
was not as general as
supposed.
We replicated our Maple
results using Mathematica,
suggesting a closer look at
Lifshitz theory.

Casimir
forces
for
non‐homogeneous
media
Further analysis gave an
extension to Lifshitz theory
that
(i) gives the same results
for previous models
(ii) gives finite integrands
in our case
We can now calculate the
Casimir force for the
exponential decay model …
… and, in principle, for any
inhomogeneous model

Casimir
forces
for
non‐homogeneous
media
The Casimir force is attractive close to the ε 1 boundary, then
becomes negative.

Challenges
for
future
research
We
have
been
opera:ng
at
the
limits
both
of
what
can
be
computed
symbolically
and
the
exis:ng
theory.
80%
symbolic
computa:on,
20%
numeric
(say).
We
know
that
we
have
divergence
at
the
boundaries
for
the
exponen:al
decay
model.
We
know
that
we
would
not
have
divergence
for
a
C ∞ 
model.
All
our
op:ons
for
C n 
models
appear
to
involve
80%
numeric
and
20%
symbolic
computa:on,
roughly
speaking.
Moreover,
Casimir
forces
are
small,
so
numeric
precision
is
vital.
And
we
will
need
to
solve
large
numbers
of
systems
of
BVP
ODE’s
(if
I
can
work
out
how
to
linearise
the
second
order
BVPs)
so
HPC,
grid,
cloud,
etc.
possibili:es
will
need
to
be
considered.

CuDng
Edges
We
are
working
on
the
boundary
of
exis:ng
theory
–
it
may
or
may
not
be
possible
to
design
experiments
that
confirm
or
refute
our
hypotheses.
We
are
working
on
the
boundary
of
the
symbolic/numeric
computa:onal
divide.
And
we
are
working
with
small
(picoNewtons)
forces,
so
the
tradeoffs
are
not
in
our
favour.
For
example,
small
effects
in
the
finite
element
analysis
of
aircraf
wing
design
are
unimportant:
will
this
large
effect
result
in
something
bad
happening?
Standard
error
analysis
is
meaningless
if
you
can’t
specify
what
an
error
is
‐
and,
at
the
moment
‐
we
have
no
idea.
We
are
working
on
the
boundary
of
sequen:al
and
grid/cloud/distributed/parallel
computa:on.
If
we
reduce
x‐points
and/or
frequency
step‐sizes,
can
we
derive
any
meaning

from
our
results?
If
we
don’t,
are
our
planned
computa:ons
tractable
on
exis:ng
architectures?

Casimir
forces:
Interes:ng,
important
and
very
poorly
understood