Quantum Electronics Seminar presentation Tobias Kippenberg

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Quantum Electronics Seminarpresentation Tobias
Kippenberg

• Topic The Casimir force

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Outline

• Historical overview
• A quantum mechanical derivation of the Casimir
  force on two parallel conducting plates
• The Casimir force in the age of nanotechnology
  MEMS
• Effects of the casimir force on MEMS
• Dynamical effects of the casimir force

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The Casimir force
Hendrik Casimir (1909-2000)
Koninkl. Ned. Akad. Wetenschap. Proc. 51, 793
(1948).
Source physicsweb.org
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Family of Casimir forces

• Several attractive forces are termed Casimir
  forces
• Attraction between an atom and a dielectric
• Attraction between an atom and a metalic surface
• Attraction between two dielectrics
• Attraction between two metallic surfaces.

Casimir-Polder forces
Casimir-forces
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Zero point energy

• Zero point energy First introduced by Planck
  second theory of black body radiation (ca. 1930)

Zero point energy was needed to reconcile
thermodynamics with radiation spectrum, when
considering a harmonic oscillator in equilibrium
with temperature.
Initially thought that the zero point Energy was
of no physical consequence.
The total zero point energy of all modes is
infinite!
e.g. P.W. Milonni, Contemporary Physics 1992, V
33, No. 5, 313
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Derivation of the Casimir forceby zero-point
energy consideration
Casimirs idea was to look at situations where
the zero point Energy might change. He considered
two parallel conducting plates.
The allowed frequencies obey
The vacuum energy is the sum of the now discrete
set of modes
He considered the differences of the vacuum energy
Closely spaced
Infinite separation
e.g. P.W. Milonni, Contemporary Physics 1992, V
33, No. 5, 313
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Derivation of the casimir forceby zero-point
energy consideration

• The sum of these two infinite sums can be
  calculated (Euler-Maclaurin series). This change
  in energy give rise to a attractive force between
  the plat

• This gives rise to an attractive force, varying
  as the fourth power between two metallic plates
  separates a distance z

• First successful experiments were reported by
  Derjaguin et al. in 1951. good quantitative
  agreement due to the lack of absorption spectra

e.g. P.W. Milonni, Contemporary Physics 1992, V
33, No. 5, 313
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Casimir-type effect in fluid dynamics
Source Buks, Roukes, newsviews, Nature 2002
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Alternative interpretation Linear momentum of
photons

• The casimir force can also be interpreted by
  considering the radiation pressure due to the
  photon momentum.

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Casimir force for different geometries

• Casimir force for two conducting plates

• 1cm x 1cm conducting plates separated by 1
  micron. Casimir Force F 130 nN
• Force is identical to the gravitational
  attraction of two blocks of mass m 1lb 0.5 inch
  apart

• Casimir force between a plate and a conductor

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Resurgence of Casimir force

• Driving force MEMS micro-electro-mechanical
  systems

Stiction
Source www.sandia.gov
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Casimir force experiment
Source physicsweb.org
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Casimir force measurement
Heavily doped poly-silicon top plate
Torsional rod (rotation axis)
Electrical contacts
archor for torsional rod
Source Capasso et al. Science
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Casimir force measurements setup

• Capacitance measurements to measure torque angle
• Application of DC voltage to cause a torque
• Use a AC voltage to sense the change in
  capacitance

?C1 part 2x105 Equivalent angle 8x10-8 rad
Source Capasso et al. Science
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Calibration using electrostatic attraction
To measure a force, the mechanical system need to
be characterized in terms of the torsional spring
constant.
Small amplitude approximation
Electrostatic attraction (V290mV)
?6x10-5 N rad-1 z0 67 nm V0 30 mV (work
function difference of two Gold surfaces)
Source Capasso, Science, Vol. 291 (2001)
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Casimir force measurements
By turning of the electrostatic attraction by
proper biasing the Casimir force can be
observed!!!

• Maximum force is ca. 700 pN
• forces become observable at ca. 300 nm
• Short range 1/r3
• Surface roughness increases Casimir force
• transmission of gold at 500 nm weakens the
  Casimir force

Electrostatic attraction
Source Capasso, Science, Vol. 291 (2001)
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Pertubation of a harmonic oscillator by the
Casimir force
Source Capasso et al. Phys. Rev. Lett. (2001)
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Casimir oscillator
2V

• The torsional rod is driven
• by an AC voltage
• Mechanical resonance frequency at 2.7KHz
• Oscillator is harmonic

V
Source Capasso et al. Phys. Rev. Lett. (2001)
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Anharmonic oscillator due to Casimir force
Shifted resonance

• Driving force (torque)
• Natural oscillation frequency
• Damping coefficient
• Casimir force
• Anharmonic contributions due to the Casimir
  force
• 2nd order
• 3rd order

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Casimir oscillator in the small oscillation regime
IN the small pertubation regime the anharmonic
terms which are cubic and quadratic in order can
be neglected.
Casimir force
Electrostatic attraction
Source Capasso et al. Phys. Rev. Lett. (2001)
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Nonlinear Casimir oscillator
3300 nm
141 nm
116.5 nm
98 nm
Source Capasso et al. Phys. Rev. Lett. (2001)
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Outlook

• - Casimir force is a non-intuitive force which
  exemplifies the profound influence of quantum
  vacuum
• - The force is likely to receive revived interest
  in the future due to the drive in miniaturization
  and high sensitivity measurems.

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Historical overview of the Casimir force

• Prediction Hendrik Casimir of Philips Research
  Laboratories in the Netherlands
• "colloidal solutions". These are viscous
  materials, such as paint and mayonnaise, that
  contain micron-sized particles in a liquid
  matrix. The properties of such solutions are
  determined by van der Waals forces - long-range,
  attractive forces that exist between neutral
  atoms and molecules.
• 1932 Theo Overbeek, realized that the theory
  that was used at the time to explain van der
  Waals forces, which had been developed by Fritz
  London in 1932, did not properly explain the
  experimental measurements on colloids. Overbeek
  therefore asked Casimir to investigate the
  problem. Working with Dirk Polder, Casimir
  discovered that the interaction between two
  neutral molecules could be correctly described
  only if the fact that light travels at a finite
  speed was taken into account
• Soon afterwards, Casimir noticed that this result
  could be interpreted in terms of vacuum
  fluctuations. He then asked himself what would
  happen if there were two mirrors - rather than
  two molecules - facing each other in a vacuum. It
  was this work that led to his famous prediction
  of an attractive force between reflecting plates.