The Casimir effect

Physics 250 Spring 2006 Dr Budker Eric Corsini Casimir Patron Saint of Poland and Lithuania (March 4 th ) Hendrik Casimir (1909-2000) Dutch theoretical physicist Predicted the “force from nowhere” in 1948

Abstract The Casimir Force

    The Casimir Force was first predicted by Dutch theoretical physicist Hendrik Casimir and was first effectively measured by Steve Lamoreaux in 1995.

The boundary conditions imposed on the electromagnetic fields by metallic surfaces lead to a spatial redistribution of the zero-point energy mode density with respect to free space, creating a spatial gradient of the zero-point energy density and hence a net force between the metals. That force is the most significant force between neutral objects for distances <100nm Because of that dependence on boundary conditions, the Casimir Force spatial dependence and sign can be controlled by tailoring the shape of the interacting surfaces.

In this presentation I briefly review the formalism pertaining to the zero point energy and summarize the recent experiment By Bell and Lucent labs, investigating the effect of the Casimir Force on a dynamic system.

Origin of the Casimir force

The short answer

 The vacuum cannot have absolute zero energy  that would violate Heisenberg uncertainty principle.

The long answer



“green” book approach

 We show a 1-1 relationship: SHO ↔

E&M

Field    Maxwell + Coulomb gauge (  .A=0)  (no local current/charge)  2     2 

t

2    0 General sol to wave equation    (

r

 ,

t

)  Then   (

r

 ,

t

)  1 (

C



e i

(

k

 .



r

 

t

)  ˆ 

C

 *

e



i

(

k

 .



r

 

t

)  ˆ * ) 1

V V

(

C

(

t

)

e i

( 

k

.

r

 )  ˆ 

C

* (

t

)

e



i

( 

k

.



r

)  ˆ * )   1 8  

V

(

E

2 

B

2 )

dV

     1 2   2 {|

c

2 Re[

C

(

t

)] | 2  | Im[

C

(

t

)] | 2 }

Consider the SHO



SHO



p

2 2

m



m

 2

q

2 2  Re 

scale

: 

p

 

m

 

P

,

q

   1

m



Q

  

SHO

  2 (

Q

2 

P

2 )   Note:

p



m dq dt

 

P



dQ



then Q

(

t

)  

o

cos( 

t

),

P

(

t

)   

o

 sin( 

t

)

dt

Re[

C

(

t

)] 

Q

(

t

) Then there is a 1-1 relation Im[

C

(

t

)] 

P

(

t

)  If we set α

o

to be such that  Then, per mode ω we have:

C

(

t

)  

E

&

M



c

2  (

Q



iP

)   2 (

Q

2 

P

2 )

We can then apply the SHO mechanics to the E&M field

  Eigenstates |n> Eigenvalues E n = ħω(n+ 1 / 2 )  In particular E o = ħω/2 ≠ 0 for mode ω  However     1   / 2  

But we are only concerned in the difference in energy density

 Between two conducting parallel plates only virtual photons whose wavelengths fit a whole number of times between the plates contribute to the vacuum energy  there is a force drawing the plates together.

F

   c 480 A d 4 A  1cm 2 , d  1  m 

F

 10 7 N or Pressure  10 3 Pascal d  10nm  Pressure  10 5 Pacal  1atm  Strongest force between tw o neutral objects (d  10nm)

Notes

  Bosons  attractive Casimir force Fermions  repulsive Casimir force  With supersymmetry there is a fermion for each Boson  no Casimir effect.

 Hence if supersymmetry exists it must be a broken symmetry

Casimir Force From theory to experiment

Steven Lamoreaux’ experimental set up   Steve Lamoreaux (University of Washington – Seattle) Measured the Casimir force between a 4 cm diameter spherical lens and an optical quartz plate about 2.5 cm across, both coated with copper and gold. The lens and plate were connected to a torsion pendulum.

   There are only a few dozen published experimental measurements of the Casimir force But there are more than 1000 theoretical papers And citations of Casimir’s 1948 paper are growing exponentially.

Effects of edges

shape of decay function is strongly dependent on size and separation of surfaces ref:http://images.google.com/imgres?imgurl=http://www.sr.bham.ac.uk/yr4pasr/project/casimir/currentthumb.jpg&imgrefurl=http://www.sr.bham.ac.uk/yr4pasr/project/casimir/&h=275&w=275&sz=41&tbnid=Buy2QDUNZEvi6M:&tbnh=109&t bnw=109&hl=en&start=20&prev=/images%3Fq%3Dcasimir%2Beffect%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DG

Dist > 25µm: dome shape The Casimir force occurs when virtual photons are restricted.

The force is reduced where virtual photons are diffracted into the gap between the plates Unshaded areas correspond to higher Casimir forces Casimir force is decreased at the edges of the plates

The Casimir force: F C on Microelectromechanical systems (MEMS) (PRL: H. B. Chan et al – Bell Lab & Lucent Tech –Published Oct 2001)

        Prior experiments have focused on static F C and adhesion F C This experiment investigates the dynamic effect of F C: A Hooke’s law spring provides the restoring force F C between a movable plate and a fixed sphere provides the anharmonic force For z>d

CRITICAL

 system is bistable PE has a local + global minima F C makes the shape of local min anharmonic Note: chosing a sphere as one of the surfaces avoids alignment problems Mock set up K= 0.019 Nm-1 Sphere radius = 100μm d

EQUILIBRIUM

= 40nm

F

   c 480 A d 4

to F

  3  c 360 R d 3

The actual set up

      Oscillator: 3.5-mm-thick, 500-mm 2, gold plated (on top), polysilicon plate Room temp – 1 milli Torr A driving voltage

V

AC excites the torsional mode of oscillation (

V

DC1 : bias) V dc: bias to one of the two electrodes under the plate to linearize the voltage dependence of the driving torque V DC2: detection electrode Note: amplitude increases with V AC = 35.4μV to 72.5 μV Torsional Spring constant: k=2.1 10-8 Nrad-1 Fund res. Freq. = 2753.47 Hz I = 7.1 10-17 kgm 2 System behaves linearly w/o sphere

Add a gold plated polystyrene sphere radius = 200μm

 Equation of motion  Freq shift ~ F C gradient (F C ’)

Ignoring

 1   0

the terms in

 2 [ 1 

b

2 2

F C



I

 0 2 ]

and

 3

F

(

z



b

 ) 

Taylor ex pand about z up to

 3     2   

F

 (

z

)   [ 

o

2 

b

2

F

 (

z

)] 

I

 3  c R 120 ( 

z



z

1 ) 4

z

1   cos 

t



dist ance



of

2  

closest

3

approach

 

o



k

,

I

  

amplitude of damping driving coef torque

  

b

3

F C

 , 2

I

  

b

4

F C

 6

I

Due to F C Due to Electrostatic force z (equil dist sph-plate w/o F C )

F

C



anharmonic behavior

    I: Sphere far away  Sphere is moved closer to plate I  Res. freq shifts as per model   1 IV   0 [ 1 

b

2 At close distance  normal resonnance hysteresis occurs 2

I F C

  0 2 ] ie: amplitude A has up to 3 roots:

A

2 [(      1 3  8  1  

A

2   2 ]  4

I

 2 2  1 2  5  12  1 3 2 

characteri zes non linearity

Freq < resonant freq Depends on history Freq > resonant freq  Or we can keep a constant excitation freq (2748Hz), vary sphere-plate distance, and measure amplitude.

Is repulsive Casimir force physical ?

 Plate-plate: attractive  Sphere-plate: attractive  Concave surface – concave surface: can be repulsive or attractive depending on separation  pendulum  Plate-plate with specific dielectric properties can be repulsive  nanotech applications

References

  

Nonlinear Micromechanical Casimir Oscillator [

PRL

:

published 31 October 2001 H. B. Chan,* V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and Federico Capasso†

Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974 Physics World article (Sept 2002) – Author:Astrid Lambrecht

REPORTS ON PROGRESS IN PHYSICS Rep. Prog. Phys.

68

(2005) 201–236

Steven Lamoreaux