Thermal effects of the Casimir forces Thermal effects of the Casimir - - PowerPoint PPT Presentation

Thermal effects of the Casimir forces Thermal effects of the Casimir forces

• n ultra
• n ultra-
• cold gases

cold gases

University of Trento University of Trento University of Trento University of Trento

International Workshop on ‘’ADVANCES IN PRECISION TESTS AND EXPERIMENTAL GRAVITATION IN SPACE’’, Galileo Galilei Institute , Florence 28-30 Sept. 2006

University of Trento, INFN and University of Trento, INFN and University of Trento, INFN and University of Trento, INFN and CNR CNR CNR CNR-

• INFM BEC Center on Bose

INFM BEC Center on Bose INFM BEC Center on Bose INFM BEC Center on Bose-

• Einstein Condensation, Trento, Italy

Einstein Condensation, Trento, Italy Einstein Condensation, Trento, Italy Einstein Condensation, Trento, Italy

• The Team

The Team

• Boyle and Gay-Lussac ideal gas lows could be explained by the

kinetic theory of non-interacting point atoms (Joule, Kroning, Clausius,..), but are hardly exact

• J.D. van der Waals (1873): eq. of state
• London (1930!): interaction potential between two atoms due to fluctuations of the

atomic electric dipole moment d dispersion forces (it is necessary only that , the vacuum is a q.s. with observable physical

consequences!)

• Casimir and Polder (1947): inclusion of retardation effect and at large

distance

( )

nRT b V v a P = −       +

2

6

1 r VVL − ∝

→ →

= ≠ = E d d d

i i

α , ,

2 7

1 r VCP − ∝

∞ ≠ c

c

r λ >>

≠ α

+ orientation forces (Keesom,T, perm. dipoles) + induction forces (Debye,q-d) = 3 types vdW forces

• nRT

PV =

• by adding the vdW force between the atoms of the two plates and assuming a

pairwise potential V=-B/r^n but this was experimentally wrong!

• the vdW force is not additive: the force between two atoms depends of the

presence of a third atom

• Lifshitz (1955), Dzyaloshinskii and Pitaevskii (1961) developed a Macroscopic

General Theory of the vdW Forces motivated by the experimental discrepancy with microscopic-additive theories

I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Advances I.E. Dzyaloshinskii, E.M. Lifshitz and L.P. Pitaevskii, Advances in Physics 38, 165 (1961). in Physics 38, 165 (1961). Heroic Era! Heroic Era!

• Lifshitz assumed the dielectrics characterized by randomly fluctuating sources

as demanded by the FDT and solved the Maxwell equations using the Green function method

• Ginzburg (1979): ''the calculations are so cumbersome that they were not even

reproduced in the relevant Landau and Lifshitz volume where, as a rule, all important calculations are given'‘

Lifshitz Lifshitz

Recent Measurements of Casimir Force Recent Measurements of Casimir Force

15 300 10-5 Cr (50nm)

• n Si

500 - 3000 Interfero- metry 2002

• G. Bressi, G.

Carugno, R. Onofrio & G. Ruoso 1 300 10-4 Cu/Au 200 - 2000 MEMS torsion bar capacitance 2003

• R. S. Decca, D.

Lopez, E. Fischbach & D. E. Krause 1 300 5x10-2 Al (300nm) + AuPd (20nm) 100 - 900 AFM 1999

• G. L. Klimthitskaya,
• A. Roy, U. Mohideen

and V. M. Mostepanenko <1 1 1 2 2 5 Accura cy (%) 20 - 1000 300 300 300 300 300 Temp (K) 10-11 Si, Au 10 - 1000 AFM, MEMS 2005

• NANOCASE

1000 Au (200nm) + Cr underlayer 90 - 1000 MEMS torsion bar capacitance 2001

• H. B. Chan, V. A.

Aksyuk, R. N. Kleiman, D. J. Bishop & F. Capasso 1000 50µm Au wires coated in thiol SAM 20 - 100 Piezo-tube manipulator 2000

• T. Ederth

5x10-2 Al (250nm)+ AuPd (8nm) 100 - 900 AFM 1999

• A. Roy and U.

Mohideen 5x10-2 Al (300nm) + AuPd (20nm) 100 - 900 AFM 1998

• U. Mohideen & A.

Roy 10-4 Au(500nm) 600 - 6000 Torsion pendulum 1997

• S. K. Lamoreaux

Pressu re (mbar) Materials Distance Scale (nm) Method Geometry Year Investigators

• Behaviour of Casimir-Polder force well explored experimentally at short

distances (mainly forces between metallic bodies) Bressi et al. PRL 2002 (plate-plate configuration)

• Behaviour at larger distances (few microns) less explored. In particular

thermal effects of the force not yet measured

• Cold atoms are natural candidates to explore thermal effects of the force

at moderately large distances (5-10 microns).

Surface Surface-

• atom interaction

atom interaction has been the has been the object

• bject of
• f systematic

systematic experimental experimental and and theoretical theoretical studies studies in in recent recent years years. .

Motivations : Motivations :

• Open theoretical and experimental questions

(ex: role of e.m. thermal fluctuations, usually masked)

• Perspectives for applications (atom chips, ..)
• New constraints on hypothetical non-Newtonian forces

at short distances

• Shih and Parsegian (1975): deflection of atomic beam (VL)
• Anderson (1988): deflection of atomic beam (VL), Rydberg atoms
• Hinds (1993): deflection of atomic beam (CP)
• Aspect (1997): reflection from atomic mirror
• Shimizu (2001, 2005): reflection from solid surface
• Vuletic (2004): BEC stability near surfaces
• Ketterle (2004): BEC reflection from solid surface
• Cornell (2005): BEC center of mass oscillation (CP)
• Cornell (2006): BEC center of mass oscillation (Thermal)

Plan of the talk Plan of the talk

• force at
• force
• Recent
• force

• )

, ( ) ( ) , ( ) ( ) , ( ) ( ) ( t r E t d t r E t d t r E t d r F

ind i fl i fl i ind i tot i tot i

• ∇

+ ∇ ≈ ∇ =

E

T

Force includes zero-point (or vacuum) fluctuations effects + thermal (or radiation) fluctuations effects (crucial at large distance!)

• S

T

ij B S fl j fl i

r r T k r P r P δ δ ω ω δ ω ω ε ω ω ) ' ( ) ' ( 2 coth 2 ) ( ' ' ] ' , ' [ ] ; [

• −

−         =

+

r P r r G r E

V

• d

] [ ] ' , ; [ ] ; [ ω ω ω

• = ∫

Electric Field Fluctuations Dissipation Theorem

!" !" # #

• )

, ( ) ( ) , ( z T F z F z T F

eq th eq

+ =

[ ]

r r r ii z B eq

r r G T k d z T F

• =

= ∞

∂         =

∫

2 1 2

] , ; [ ) ( Im 2 coth ) , (

2 1

ω ω α ω ω π

1 2 1 2 coth

/

− + =        

T k B

B

e T k

ω

ω

• Thermal fluctuations

Vacuum fluctuations : T=0

• Optical length

fixed by optical properties of the substrate ( typically fractions of microns)

• Thermal photon wavelength ( at room temperature)
• pt

λ m T k c

B T

µ λ 6 . 7 / ≈ =

van der Waals-London (Vacuum) Casimir-Polder (Vacuum+retardation) Lifshitz (Thermal)

• $ !

$ !

• Only static optical properties

characterize the asymptotic behaviour of Casimir-Polder and thermal (Lifshitz) forces

• At smaller distances (van der Waals regime) dynamical optical properties

end are needed

) ( ) 1 ( ) 1 ( 2 3 ) ( ) (

5

ε φ ε ε π α λ λ + − − = → << << z c z F z F z

CP T

• pt
• )

1 ( ) 1 ( 4 3 ) , ( ) , (

4

+ − − = → >> ε ε α λ z kT z T F z T F z

Lif eq T

% %! ! ! !

) ( ' ' ) ( ' ) ( ω α ω α ω α i + =

ξ ξ ε ξ ε ξ α π λ d i i i z z F z F z

VL

• pt

1 ) ( 1 ) ( ) ( 4 3 ) ( ) (

4

+ − − = → <<

∫

∞

• ω

ξ ω ω α ω π ξ α ω ξ ω ω ε ω π ξ ε d i d i

2 2 ' ' 2 2 ' '

) ( 2 ) ( ) ( 2 1 ) ( + = + + =

∫ ∫

∞ ∞

) ( ' ' ) ( ' ) ( ω ε ω ε ω ε i + =

) , (

0 ε

α increases linealy with T

van der Waals-London Casimir-Polder Lifshitz (thermal)

Surface (sapphire) atom (rubidium) interaction at T=300K

[ M. Antezza, L.P. Pitaevskii, S.Stringari, Phys.Rev A70, 053619 (2004)]

• Surface-atom force extremely weak at large distances

(typically 10E-4 gravity at 4-5 microns)

• At room temperature thermal effects prevail only above 5-6 microns

and are consequently difficult to measure

• Possible

Possible strategies strategies: :

• increase T

(thermal effect increases linearly with T, but vacuum in the chamber?)

• out of thermal equilibrium configurations

(if surface is hotter than environnment thermal effect increases quadratically with surface temperature)

• Casimir-Polder force already

detected in various experiments

• How to detect thermal effects ?

• Thermal effect in surface-atom force can be tunable by varying substrate

and environment temperatures.

• What happens if substrate and environment temperatures are different ?
• How to describe environment radiation and to calculate field average values?
• !

!

Or viceversa: cold surface and hot environment

• C. Henkel, K. Joulain, J.-P. Mulet and J.-J. Greffet, J. Opt. A 4,S109 (2002)
• M. Antezza, L.P. Pitaevskii and S. Stringari, PRL 95, 113202 (2005)

S

T

E

T

short distances medium and long distance behaviour

10

12

10

13

10

14

10

15

10

16

10

17

50 100 150 200 250 300

ω (rad/sec) ε‘‘(ω) Experimental data of ε‘‘(ω)

10

12

10

13

10

14

10

15

10

16

10

17

−100 −80 −60 −40 −20 20 40 60 80 100

ω (rad/sec) ε‘(ω) Experimental data of ε‘(ω)

) ( ' ' ) ( ' ) ( ω ε ω ε ω ε i + =

K

2 1

10 6⋅ ≈ ω

# #%! %!

• Sapphire (Al2O3) substrate

Rubidium ( Rb) atoms

2.3 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.4 x 10

−0.5 0.5 1 1.5 2 2.5 x 10

ω (rad/sec) Re α(ω)

Real part of the α(ω) function for a 87Rb atom

) ( ' ' ) ( ' ) ( ω α ω α ω α i + =

K

2 2

10 9⋅ ≈ ω

K

at 4

10 2⋅ ≈ ω

87

% %

) , ( z T Feq

th

1 ] , ; [ Im ) ( ' 2 ) , (

/ 2 1 ,

2 1 2

− ∂ =

= = ∞

∫

T k r r r ii z ff eq th

B

e r r G d z T F

ω

ω ω α ω π

• 1

] , ; [ Re ) ( ' ' 2 ) , (

/ 2 1 ,

2 1 2

− ∂ =

= = ∞

∫

T k r r r ii z df eq th

B

e r r G d z T F

ω

ω ω α ω π

• )

( ) ( ' '

at

ω ω δ ω α − ≈

at BT

k ω

• <<

⇒

) , ( ) , (

,

z T F z T F

ff eq th eq th

≅

Field fluctuations provide leading term also out of thermal equilibrium at E B S B

T k T k ω

• <<

,

• Atom does not contribute to thermal radiation!
• Thermal

component of the force is determined by Stark effect

th z th

E F > < ∂ =

2

2 1α

1 1 ) ( 6

3 2 2 2

− + − − = ε ε α π

• c

z T T k F

E S B neq

• force decays slower than at thermal equilibrium:
• force depends on temperature more strongly than at equilibrium
• force can be attractive or repulsive depending on relative

temperatures of substrate and environment

• force has quantum nature
• simple extension to metals (Drude model )

) 1 ( 4 ) 1 ( 3

4

+ − − = ε ε α z T k F

B eq

• substrate

environment

) , , ( ) , , ( ) , ( ) , , ( z T F z T F z T F z T T F

E th S th E eq E S

− + =

holds at low temperature

ω πσ ε / 4

' ' =

• M. Antezza, L.P.Pitaevskii and S.Stringari, Phys. Rev. Lett. 95,093202 (2005)

• T=0K

T=300K

• Sapphire substrate
• Rubidium atoms

T=600K

3 4 5 6 7 8 9 10 −12 −10 −8 −6 −4 −2 2 x 10

−23

z [µm] F(TS,TE,z)=−∂z V(TS,TE,z) [dyne] 3 4 5 6 7 8 9 10 −12 −10 −8 −6 −4 −2 2 x 10

−23

z [µm] F(TS,TE,z)=−∂z V(TS,TE,z) [dyne] 3 4 5 6 7 8 9 10 −12 −10 −8 −6 −4 −2 2 x 10

−23

z [µm] F(TS,TE,z)=−∂z V(TS,TE,z) [dyne] 3 4 5 6 7 8 9 10 −12 −10 −8 −6 −4 −2 2 x 10

−23

z [µm] F(TS,TE,z)=−∂z V(TS,TE,z) [dyne]

Non-equilibrium: substrate T=300K environment T=600K Non-equilibrium: substrate T=600K environment T=300K

equilibrium

• Availability of Bose-Einstein condensates and degenerate Fermi gases

yields new perspectives in the study of surface-atom forces !&%

• scillations+interference
• Collective oscillations with BEC’s: first experiment at JILA (2005)

(sensitive to the gradient of the force)

• Bloch oscillations with ultracold degenerate gases

(sensitive to the force)

• Macroscopic BEC phase interference in double well potentials

(sensitive to the potential) Bose-Einstein-condensed gases are dilute, ultracold samples characterized by unique properties of coherence and superfluidity. They give rise, among others, to a variety of collective oscillations (S. Stringari (1996))

Attractive force -> Trap frequency decrease Unperturbed trap, ω Modified trap, ω Move near the surface Use trapped BEC as a mechanical oscillator: Measure changes in oscillation frequency Oscillating BEC Surface

Measuring atom-surface interactions: dipolar oscillations of a BEC

r d z V r n m a r d z V r n m

at surf z at surf z z cm

• )

( ) ( 8 ) ( ) ( 1

4 2 2 2 2 − −

∫ ∫

∂ + ∂ = −ω ω

In M. Antezza, L.P. Pitaevskii and S. Stringari, PRA 70, 053619 (2004), the surface-atom force has been calculated and used to predict the frequency shift of the center of mass

• scillation of a trapped Bose-Eisntein condensate, including:
• Effects of finite size of the condensate
• Non armonic effects due to the finite amplitude of the oscillations
• Dipole (center of mass) and quadrupole (long lived mode) frequency shifts

' ' ($ ($

2 2 2 2 2 2

2 2 2 ) ( z m y m x m r V

z y x ho

ω ω ω + =

• In the presence of harmonic potential

+ surface-atom force frequency of center of mass motion is given by Linear approximation First non-linear correction

) cos( t a Z Zcm ω + = ≡ ) (

0 r

n

a= amplitude of c.m. oscillation Thomas-Fermi inverted parabola

2 4 1002 1004 1006 325 300 275 250 225

Expanded position (µm) Oscillation Time (ms)

Surface 1) Make BEC far from surface 2) Push BEC a few microns from surface 3) Excite oscillation vertically 4) Switch to anti-trapped state (atoms fall) 5) Image atoms on CCD camera

• Measuring atom-surface interactions: dipolar oscillations of a BEC

• Multiple dielectric

surfaces! Amorphous glass, crystalline sapphire.

• No conducting objects

near atoms!

• Can sustain high

temperatures and be compatible with UHV!)

• The experimental apparatus

Exp: D.M. Harber, J.M. Obrecht, J.M. McGuirk ,and E.A. Cornell, PRA 72, 033610 (2005)

• )*

)* +, +,

• (

( $ $

• Thermal effects not

yet evident some months ago! Frequency shifts strongly enhanced by non equilibrium effects !?!

• Fused Silica substrate
• Rubidium atoms
• Experiment at room temperature

van der Waals

• --- 500 K
• --- 300 K Theory: Antezza, Pitaevskii, Stringari, Phys. Rev A 70, 053619 (2004)
• 0 K

Recent Experimental results from JILA

7 8 9 10 11 1 2 3 4 5

310K 474K 605K Room Temperature Environment Surface Temperature: Normalized Frequency Shift (10

• 4)

Trap Center - Surface Distance (µm)

!"! !"!# #$! !%$"! $! !%$"!&& &&$'!(! $'!(!() ()$! $!* *$!!+$,--.$ $!!+$,--.$/0 /01 10608074 0608074

100 200 300 400 500 600 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Non-Equilibrium: TENV = 310K Equilibrium: TSURF = TENV Casimir - Polder: TSURF = TENV = 0K FFS (10

• 4)

Surface Temperature (K)

• Experiment on collective oscillation probes gradient of the force
• Due to finite size of condensate and amplitude of oscillation

experiment does not probe the effects locally (average sensitive to shorter distances where thermal effects are weaker). ("- ("-

• %

%

• Sensitive measurement of forces at micron scale using Bloch osci

Sensitive measurement of forces at micron scale using Bloch oscillations llations I.

• I. Carusotto

Carusotto, L. , L. Pitaevskii Pitaevskii, S. , S. Stringari Stringari, G. , G. Modugno Modugno, and M. , and M. Inguscio Inguscio, ,

• Phys. Rev. Lett. 95, 093202 (2005)
• Center of mass oscillation
• measures gradient of the force
• mechanical approach (oscillation in coordinate space)
• Bloch oscillation
• measures directly the force
• interferometric approach (oscillation in momentum space)

• atomic gas initially feels

3D harmonic trap+ gravity + periodic confinement

• at t=0 one switches off

harmonic trap System feels periodic potential + gravity and starts oscillating (Bloch oscillation).

• After given evolution time the

periodic potential is switched off. Atomic gas falls down, expands and is hence imaged.

• For ideal gas imaged profiles

are proportional to initial momentum distribution

( (

• 2

/ λ ω mg

Bloch =

Atoms filling different wells evolve with different phase due to gravity ! (interferometric tool)

mg D FCP

4

10 / ≈

CP g F

F

Bloch oscillations of a trapped gas in an optical lattice in presence of gravity and surface-atom interactions: change in the ext. force change in the Bloch frequency Sensitivity required: ∆TB/TB = 10-4 -10-5

• q
• = F = mg+ F

CP

5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 z [µm] ∆ TB/TB (x 10−4) ∆ TB/TB for the full force but with the single point approximation

• (

(

T=0K T=300K T=600K

Non-equilibrium: substrate T=600K environment T=300K

Carusotto, Antezza, Pitaevskii, Stringari

Theoretical prediction for trapped K fermi gas

• .

. $&% $&% (/ (/

• M. Ben Dahan et al., Phys. Rev. Lett. 76, 4508 (1996)
• E. Peik et al., Phys. Rev. A 55, 2989 (1997)
• R. Battesti et al., Phys. Rev. Lett. 92, 253001 (2004)
• G. Roati

et al., Phys. Rev. Lett. 92, 230402 (2004)

• P. Lemonde, and P. Wolf, Phys. Rev. A 72, 033409 (2005)
• G. Ferrari et al., Phys. Rev. Lett. 97, 060402 (2006)

• (

)

t r T l T P

zz neq

, ) , (

• =
• Role of thermal fluctuation?
• is it possible to recover

the surface-atom force?

• what for asymptotics?
• Dorofeyev, J. Phys. A: Math. Gen. 31, 4369 (1998) –

equal materials

• Dorofeyev et al., Phys. Rev. E 65, 026610 (2002) –

different materials

2 ) , ( 2 ) , ( ) , , (

2 1 2 1

l T P l T P l T T P

eq th eq th neq th

+ =

) , ( 2 ) , ( ) , , ( ) , ( 2 ) , ( ) , , ( l T P l T P l T P l T P l T P l T P

th eq th neq th th eq th neq th

∆ − = ∆ + =

/ /

• )

, , ( ) , , ( ) , , (

2 1 2 1

l T P l T P l T T P

neq th neq th neq th

+ =

"!&&$'!(!()$!*$2!3!$/01-.-4,-5,--.

% % ! !

• %

%

Equilibrium Non Equilibrium

% % ! !

• !%

!%

#0 *0

( )

1 at holding 1 1 1 ) , , ( : 1 then and fixed with first

20 20 10 10 3 20 20

− >> − − − = → − ∞ → ε λ ε ε ε ε ε

T B neq th

C T k T P

• (

)

1 1 1 16 ) , ( for : m equilibriu At

20 10 10 3

− + − = >> ε ε ε π λ

• T

k T P

B eq th T

α π ε 4 1

20

n = −

T k c

B T

• =

λ

1 - thermal dependence 2 - distance dependence 3 - non-additivity 4 - ranges of validity

( ) ( )

1 at holding 1 1 1 c 24 ) ( ) , , ( ) equal! are EW and PW . . ( : then and fixed with 1 first

20 20 10 10 2 2 20

− >> >> − − − = ∞ → → − ε λ λ ε ε ε ε

T T B neq th

T k T P b n

• T

l x λ ε / 1

20 −

=

X=1

Non-additive, T/l^3 Additive, T^2/l^2

• +

+ ( )

1 1 1 c 24 ) ( ) , , (

20 10 10 2 2

− − − = ε ε ε

• T

k T P

B neq th

From the surface-rarefied body: But the surface-atom force is: ( )

1 1 1 c 24 ) ( ) , , (

20 10 10 3 2

− − − = ε ε ε

• T

k T F

B neq th

What is the problem?? If the gas occupies a finite slab L and does not absorb the thermal radiation: the inclusion of the remote surface results in a PW contribution of the

• rder

and hence should be omitted!

' ' 2 2 / ε

λ

• T

L <<

( )

3 20

1 − ∝ ε

• Surface-atom force out of thermal equilibrium exhibits new asymptotic (large

distance) behaviour and can provide a new way to mesure thermal effects

• Center of mass oscillation of a trapped Bose-Einstein condensate provides

powerful mechanical tool to detect surface-atom force at large distances, agree with theoretical predictions for Casimir-Polder force (first measurement of any thermal effect) (Trento (Trento (Trento (Trento-

• Boulder collaboration)

Boulder collaboration) Boulder collaboration) Boulder collaboration)

• Study

Study Study Study of the

• f the
• f the
• f the surface

surface surface surface-

• surface force

surface force surface force surface force out of thermal

• ut of thermal
• ut of thermal
• ut of thermal equilibrium

equilibrium equilibrium equilibrium and and and and asymtotic asymtotic asymtotic asymtotic non non non non-

• additivity

additivity additivity additivity

3 2 2

) ( ) , , ( z T T z T T F

E S E S

− →

310K 474K 605K Room Temperature Environment Surface Temperature: Normalized Frequency Shift (10

• 4)

Trap Center - Surface Distance (µm)