The Casimir effect Physics 250 Spring 2006 Dr Budker Eric Corsini Casimir Patron Saint of Poland and Lithuania (March 4 th ) Hendrik Casimir ( ) 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 The Casimir Force was first predicted by Dutch theoretical physicist Hendrik Casimir and was first effectively measured by Steve Lamoreaux in 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 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. 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. 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 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 We show a 1-1 relationship: SHO ↔ E&M Field Maxwell + Coulomb gauge ( .A=0)  Maxwell + Coulomb gauge ( .A=0)  (no local current/charge) (no local current/charge) General sol to wave equation  General sol to wave equation  Then Then

Consider the SHO Note: Note: Then there is a 1-1 relation Then there is a 1-1 relation If we set α o to be such that If we set α o to be such that Then, per mode ω we have: Then, per mode ω we have:

We can then apply the SHO mechanics to the E&M field Eigenstates |n> Eigenstates |n> Eigenvalues E n = ħω(n+ 1 / 2 ) Eigenvalues E n = ħω(n+ 1 / 2 ) In particular E o = ħω/2 ≠ 0 for mode ω In particular E o = ħω/2 ≠ 0 for mode ω However However

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. 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.

Notes Bosons  attractive Casimir force Bosons  attractive Casimir force Fermions  repulsive Casimir force Fermions  repulsive Casimir force With supersymmetry there is a fermion for each Boson  no Casimir effect. With supersymmetry there is a fermion for each Boson  no Casimir effect. Hence if supersymmetry exists it must be a broken symmetry Hence if supersymmetry exists it must be a broken symmetry

Casimir Force From theory to experiment Predicted by Dutch physicist Hendrick Casimir in Predicted by Dutch physicist Hendrick Casimir in First attempt to measure the Casimir Force: 1958 by M.J.Sparnaay First attempt to measure the Casimir Force: 1958 by M.J.Sparnaay - Used the attraction between a pair of parallel plates. - But irreducible systematic errors  measurements had a 100% uncertainty, (but it fit the expectations) Sparnaay gave three guidelines; Sparnaay gave three guidelines; - The plates should be free of any dust or debris, with as little surface roughness as possible - Static electrical charges should be removed (electrostatic force can easily swamp the weak Casimir attraction). - The plates should not have different surface potentials - Ref: "Measurements of Attractive Forces Between Flat Plates“ (Sparnaay, 1958) Physica, nd attempt and first successful results: 1996 by Steven Lamoreaux: - In agreement with theory to within uncertainty of 5%. 2nd attempt and first successful results: 1996 by Steven Lamoreaux: - In agreement with theory to within uncertainty of 5%. Several other successful experiments since. Several other successful experiments since.

Steve Lamoreaux (University of Washington – Seattle) 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. 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. Steven Lamoreaux’ experimental set up

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: 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

Prior experiments have focused on static F C and adhesion F C Prior experiments have focused on static F C and adhesion F C This experiment investigates the dynamic effect of F C: This experiment investigates the dynamic effect of F C: A Hooke’s law spring provides the restoring force A Hooke’s law spring provides the restoring force F C between a movable plate and a fixed sphere provides the anharmonic force F C between a movable plate and a fixed sphere provides the anharmonic force For z>d CRITICAL  system is bistable For z>d CRITICAL  system is bistable PE has a local + global minima PE has a local + global minima F C makes the shape of local min anharmonic F C makes the shape of local min anharmonic Note: chosing a sphere as one of the surfaces avoids alignment problems Note: chosing a sphere as one of the surfaces avoids alignment problems The Casimir force: F C on Microelectromechanical systems (MEMS) (PRL: H. B. Chan et al – Bell Lab & Lucent Tech –Published Oct 2001) Mock set up K= Nm-1 Sphere radius = 100μm d EQUILIBRIUM = 40nm

The actual set up Oscillator: 3.5-mm-thick, 500-mm 2, gold plated (on top), polysilicon plate Oscillator: 3.5-mm-thick, 500-mm 2, gold plated (on top), polysilicon plate Room temp – 1 milli Torr Room temp – 1 milli Torr A driving voltage V AC excites the torsional mode of oscillation 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 dc: bias to one of the two electrodes under the plate to linearize the voltage dependence of the driving torque V DC2: detection electrode V DC2: detection electrode Note: amplitude increases with V AC = 35.4μV to 72.5 μV Note: amplitude increases with V AC = 35.4μV to 72.5 μV Torsional Spring constant: k= Nrad-1 Fund res. Freq. = Hz I = kgm 2 System behaves linearly w/o sphere

Add a gold plated polystyrene sphere radius = 200μm Equation of motion  Equation of motion  Freq shift ~ F C gradient (F C ’) z (equil dist sph-plate w/o F C ) Due to F C Due to Electrostatic force

F C  anharmonic behavior I: Sphere far away  normal resonnance I: Sphere far away  normal resonnance Sphere is moved closer to plate I  IV Sphere is moved closer to plate I  IV Res. freq shifts as per model  Res. freq shifts as per model  At close distance  hysteresis occurs At close distance  hysteresis occurs ie: amplitude A has up to 3 roots:

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

Is repulsive Casimir force physical ? Plate-plate: attractive Plate-plate: attractive Sphere-plate: attractive Sphere-plate: attractive Concave surface – concave surface: can be repulsive or attractive depending on separation  pendulum Concave surface – concave surface: can be repulsive or attractive depending on separation  pendulum Plate-plate with specific dielectric properties can be repulsive  nanotech applications 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 Physics World article (Sept 2002) – Author:Astrid Lambrecht REPORTS ON PROGRESS IN PHYSICS Rep. Prog. Phys. 68 (2005) 201–236 Steven Lamoreaux