SCATTERING APPROACH FOR MOMENTUM AND HEAT TRANSFER

In collaboration with:R.L. Jaffe, M. Kardar, M. Krüger, M. Maghrebi, J. Rahi, A. Shpunt (MIT),

G. Bimonte (Napoli),N. Graham (Middlebury),

U. Mohideen, R. Zandi, E. Noruzifar (UC Riverside),

Thorsten EmigCNRS & Université Paris Sud

Nanoscale Radiative Heat Transfer, Les Houches, 16/05/2013

OUTLINE• Equilibrium fluctuations:

• interactions between macroscopic bodies

• influenced by shape, material properties, and temperature

• correlated !

• Non-equilibrium fluctuations:

• bodies at different temperatures

• novel interaction effects

• heat radiation and transfer

• Outlook, new directions

HEAT RADIATION AND TRANSFER

• Breaking the law, at the nanoscale [MITnews, July 29, 2009]

• Planck’s law is modified for small objects and short separations

• Probing Planck’s Law with Incandescent Light Emission from a Single Carbon Nanotube [Y. Fan, S.B. Singer, R. Bergstrom, & B.C. Regan, Phys. Rev. Lett.102, 187402 (2009)]

• Probing Planck's Law for an Object Thinner than the Thermal Wavelength [C. Wuttke and A. Rauschenbeutel, arXiv:1209.0536 [quant-ph]]

SURFACE PHONON POLARITONS MEDIATED ENERGY TRANSFER BETWEEN NANOSCALE GAPS

• Beyond Stefan-Boltzmann law

• Understand heat transfer in nano-systems

• Near field effects can give huge enhancement of transfer(tunneling of evanescent waves)

S. Shen, A. Narayanaswamy, & G. Chen, Nano Lett. 9, 2909 (2009)

EQUILIBRIUM QED: INTERACTIONS

• Start from path integral for free energy of electromagnetic fieldat inverse temperature and imaginary frequency

Z =⇧

�

⌃DADA� exp

⇤��

⌃dxE�(⇥,x)

�H0(⇥) +

1⇥2

V(⇥,x)⇥

E(⇥,x)⌅

F (�) = � 1�

log Z(�)

• Free photons:

• Interaction:

1/� ⇥ = ic�

V(⇥,x) = I ⇥2 (�(ic⇥,x)� 1) + �⇥�

1µ(ic⇥,x)

� 1⇥

�⇥

H0(�) = I +1�2

����

• In terms of current fluctuations:

�J(x�)

⇥G0(⇥,x,x⇥) + V�1(⇥,x)�(3)(x� x⇥)Z ⇠

ZDJDJ

⇤|obj

exp

��

Zdxdx0

J

⇤(x)

⇣

THE T-OPERATOR AND ITS DECOMPOSITIONG0(⇥,x,x⇥) + V�1(⇥,x)�(3)(x� x⇥)T�1 =

is inverse T-operator: induced source = T-operator × applied field”

E =⇥c

2⇥

� ⇥

0d� log det(MM�1

⇥ )

M =

0

@(T1)�1 U12 U13 · · ·

U21 (T2)�1 U23 · · ·· · · · · · · · · · · ·

1

A M�11 =

0

@T1 0 0 · · ·0 T2 0 · · ·

· · · · · · · · · · · ·

1

A

1

2

U↵� couples induced sources on different objects

INTERMEZZO: OPERATOR FORMALISM• Simple electrostatic example: Two metallic objects at fixed potentials

• Find potential U and surfacecharges from T-operators

• Conditions:

• Surface charges:

• Potential:

T1

T2

�↵ G0

T↵ =1

2(S↵ � 1)

U = G2�1

U = U0 +G1�2

U = G0(�1 + �2)

G↵ = G0 �G0T↵G0

U = U0 U = 0

�1 = G�10 (1�G0T1G0T2)

�1U0 = G�10

1X

n=1

(G0T1G0T2)nU0

�2 = �T2

1X

n=0

(G0T1G0T2)nU0

U = (1�G0T2)1X

n=0

(G0T1G0T2)nU0

multiple “scattering” & multipole expansion

CASIMIR POTENTIAL• The Casimir energy can be expressed as

with

• Diagonal: Scattering amplitudes (T-matrix elements) of individual objects. They describe shape and material properties.

• Off-diagonal: Translation matrices. They are universal (depend only on dimension of space and type of field) and describe relative position of objects, i.e., geometry.

• For two objects:

E =⇥c

2⇥

� ⇥

0d� log det(MM�1

⇥ )

E2 =⇥c

2⇥

� �

0d� ln det(1� N12)

M =

0

@(T1)�1 U12 U13 · · ·U21 (T2)�1 U23 · · ·· · · · · · · · · · · ·

1

A M�11 =

0

@T1 0 0 · · ·0 T2 0 · · ·· · · · · · · · · · · ·

1

A

N12 = T1U12T2U21

INTERPRETATION FOR COMPACT OBJECTS

• 2 Objects: Expansion in number (2p) of scatterings

• For each scattering consider l partial waves

• Expand scattering amplitude in

• For - Series of Casimir energy up to order

one needs only finite p, l

• Scale-free objects (cone, wedge, …): low order expansion accurate

1/L

E2 =⇥c

2⇥

� �

0d� ln det(1� N12) =

⇥c

2⇥

� �

0d�Tr log(1� N12)

� p l7,8 1 1 6

9,10 1 2 16

11,12 1 3 30

13,14 2 4 48

15,16 2 5 70

dim(N)R ⇠ R/L

⇠ (R/L)⌘

= � ~c2⇡

Z 1

0dTr

✓N12 +

1

2N2

12 +1

3N3

12 + . . .+1

pNp

12 + . . .

◆

with N12 = T1U12T2U21 ⇠ e�2L

SOME EXAMPLES

EQUILIBRIUM INTERACTIONS BETWEEN...

SHARP SHAPED METALS

T depends on R

T independent of

M. F. Maghrebi, S. J.Rahi, T. Emig, N. Graham, R. L. Jaffe, M. Kardar, PNAS 108, 6867 (2011).

PLATE - WEDGEMultiple scattering expansion is highly accurate: analytical resultsat all distances

PLATE - CONE

E = � ln 4� 116⇡

~c

d

1| ln ✓0/2| + O(✓2

0)

0 0

T=300º KT=80º K

T=0º K d = 1µm

EM

DN

�

F ⇠ �~c16⇡| ln ✓0

2 |

ln 4� 1

d2� 2

3�2T

ln2d

�T+0.810

�2T

+ · · ·�

STABLE EQUILIBRIUM?• Earnshaw’s theorem: A charged body cannot

be held in stable equilibrium by electrostatic forces from other charged bodies.

• Extension to fluctuation-induced forces?

• Start from scattering formulation (T-operators):

• Move object A by d with the “rest” of objects (R) fixed.

• Object A is unstable ( ) if

• Stability not possible for (i) (ii)on the imaginary frequency axis (where always )

E =

~c

2⇡

Z 1

0d tr log T�1T1 =

~c

2⇡

Z 1

0d tr log(I� TAGTRG)

r2d E

��d=0

0 sign(TA)sign(TR) � 0

✏J/✏M > 1, µJ/µM 1 (positive TJ)

✏J/✏M < 1, µJ/µM � 1 (negative TJ)✏J > 1

✏J , µJ

S. J. Rahi, M. Kardar, T. Emig, Phys. Rev. Lett. 105, 070404 (2010).

NON-EQUILIBRIUM QED• Objects at different temperatures (local equilibrium)

• Environment can have different temperature

• Modification of equilibrium force ?

• Radiation and transfer of heat ?

T↵

Tenv

Tenv

T1

T2

T3

T4

Parallel plates: M. Antezza, L.P. Pitaevskii, S. Stringari, V.B. Svetovoy, Phys. Rev. A 77, 022901 (2008).General shapes: M. Krüger, T. Emig, G. Bimonte and M. Kardar, EPL 95 21002 (2011), M. Antezza et al. (2011).

M. Krüger, T. Emig and M. Kardar, PRL 106, 210404 (2011).

FLUCTUATION-DISSIPATION THEOREM

• Equilibrium field correlations:

• Three contributions: ‣ zero point fluctuations: ‣ thermal currents in object :‣ environment fluctuations

Ceq(T ) = hE(!; r)E⇤(!; r0)ieq = [aT (!) + a0(!)]c2

!2ImG(!; r, r0) = C0 +

X

↵

Csc↵ (T ) + Cenv(T )

↵

C0 = a0(!)c2

!2ImG

Csc↵ (T ) = aT (!)GIm"↵G⇤

Cenv(T ) = �aT (!)c2

!2GImG�1

0 G⇤

aT (!) ⌘ !4~(4⇡)

2

c4(exp[~!/kBT ]� 1)

�1 a0(!) ⌘ !4~(4⇡)2

2c4

↵

Cneq(Tenv, {T↵}) = C0 +X

↵

Csc↵ (T↵) + Cenv(Tenv) = Ceq(Tenv) +

X

↵

[Csc↵ (T↵)� Csc

↵ (Tenv)]

C↵(T↵) ⌘ aT↵(!)G↵Im"↵G⇤↵

Csc↵ (T↵) = O↵,� C↵(T↵) O†

↵,� , with

O↵,� = (1�G0T�)1

1�G0T↵G0T�

• Non-equilibrium correlations: change temperatures

• Scattering theory: radiation of object :scattered at all other objects:

HEAT RADIATION OF SINGLE OBJECT

• Poynting vector:

• Heat emitted by object :

• Use to get general result

• Since involves only propagating waves, in matrix notation:

↵

H↵ = Re

I

⌃↵

S · n↵ = �Z 1

�1

d!

2⇡

Z

V↵

d3r hE(r) · J⇤(r)i

S(r) =c

4⇡

Zd!

2⇡hE(r)⇥B⇤(r)i

H↵ =

2~⇡

Z 1

0d!

!

exp(~!/kBT )� 1

Tr {Im[G0] Im[T]� Im[G0]T Im[G0]T⇤}

E = 4⇡i!

c2G0 J

Im[G0]

H↵ =~2⇡

Zd!

!

e~!

kBT � 1Trpr

⇥I � SS†⇤ � 0

HEAT TRANSFER BETWEEN TWO BODIES

• Two bodies, at T1 and at T2 , in cold environment. Total heat transferred from 1 to 2:

where is radiation of 1, partly absorbed by 2.

• We get for transfer rate

with

• Since J is symmetric (trace is cyclic), one gets

Htot

= H(2)

1

(T1

)�H(1)

2

(T2

)

H(2)1 (T1)

H(2)1 (T1) =

2~⇡

Z 1

0d!

!

e~!

kBT1 � 1J(T1,T2)

J(T1,T2) = Tr

⇢[Im[T2]� T⇤

2Im[G0]T2]1

1�G0T1G0T2G0 [Im[T1]� T1Im[G0]T⇤

1]G⇤0

1

1� T⇤2G⇤

0T⇤1G⇤

0

�� 0

Htot

=2~⇡

Z 1

0

d!!

1

e~!

kBT1 � 1� 1

e~!

kBT2 � 1

!J(T

1

,T2

)

TOTAL ABSORBED HEAT

• Total heat absorbed by one object (1) in the presence of a second object (2) and the environment:

• Here we have included the heat emitted by object 1 (in the presence of object 2) which is negative,

and the radiation absorbed from the environment.

• Important for heating or cooling rate.

H(1)1 = �2~

⇡

Z 1

0d!

!

e~!

kBT1 � 1ImTr

⇢(1 +G0T2)

1

1�G0T1G0T2G0 [Im[T1]� T1Im[G0]T⇤

1]1

1�G⇤0T⇤

2G⇤0T⇤

1

�

H(1)(T1, T2, Tenv) = H(1)2 (T1) +H

(1)1 (T2) +H(1)

env(Tenv)

=X

↵=1,2

H(1)↵ (T↵)�H(1)

↵ (Tenv)

NON-EQUILIBRIUM FORCE

• Maxwell stress tensor:

• Total force on object 2 due to other objects and environment:

�ab(r) =

Zd!

16⇡3

⌧EaE

⇤b +BaB

⇤b � 1

2

�|E|2 + |B|2

��ab

�

2 211F2

1 F22⌃2 ⌃2

n2 n2

F2 = Re

I

⌃2

� · n2 =

Z 1

�1

d!

2⇡

1

!

Z

V2

d3r ImhrE(r) · J⇤i = F2,eq(Tenv) +X

�

⇥F2

�(T�)� F2�(Tenv)

⇤

F(2)1 =

2~⇡

Z 1

0d!

1

e~!

kBT1 � 1<Tr

⇢r(1 +G0T2)

1

1�G0T1G0T2G0 [=[T1]� T1=[G0]T⇤

1]G⇤0

1

1� T⇤2G⇤

0T⇤1G⇤

0

T⇤2

�

F(2)2 =

2~⇡

Z 1

0d!

1

e~!

kBT2 � 1<Tr

⇢r(1 +G0T1)

1

1�G0T2G0T1G0 [=[T2]� T2=[G0]T⇤

2]1

1�G⇤0T⇤

1G⇤0T⇤

2

�

EQUILIBRIUM VS. NON-EQUILIBRIUM

• All quantities expressed as traces over product of free Green’s function and T-operators of individual bodies.

• Equilibrium:

• Computations on imaginary frequency axis

• Non-equilibrium:

• Traces are non-analytic function of frequency, computations on real frequency axis

• Quantities sensitive to details of dielectric function: resonances

• Much richer phenomenology

NON-EQUILIBRIUM EFFECTS FOR...• Heat radiation:

• Heat transfer:

• Forces

A. P. McCauley, M. T. H. Reid, M. Krüger and S. G. Johnson, Phys. Rev. B 85, 165104 (2012).

M. Krüger, T. Emig and M. Kardar, PRL 106, 210404 (2011)

M. Krüger, T. Emig, G. Bimonte and M. Kardar, EPL 95 21002 (2011)

HEAT RADIATION• Stefan-Boltzmann law for an ideal black body with surface area A:

• Sphere and cylinder (SiO2) at T=300K:

10-2

10-1

1

10-1 1 10 102

Rad

iate

d H

eat [�

T4 A

]

R [µm]

Class.

� R �t�

spherecylindercylind. ||cylind. 2 R

2 R

plate

H = �T 4A � =⇡2k4

B

60~3c2

~ volume~ surface

polarized radiation

Polarization exp. observed: Y. Öhman, Nature 192, 254 (1961);G. Bimonte et al., New J. Phys. 11, 033014 (2009).

HEAT TRANSFER• Heat transfer rate from plate to sphere (SiO2, R=5µm)

0.5

0.55

0.6

0.65

0.7

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Hea

t Tra

nsfe

r Hs [�

T4 2

�R2 ]

d / R

d = �full solutionone reflection

0.8 1

1.2 1.4 1.6 1.8

10-2 10-1 100

PTA

Rat

iod / R

10-2

10-1

100

10-1 100 101 102

Hs (

d = �

)

R [µm]

� R

Class.

2 Rd

Tenv = 0K

T = 300K

T = 0K

• Increased heat transfer at small d due to tunneling of evanesc. waves.• At small d proximity transfer approximation (PTA) is valid:• Volume-to-surface crossover around R ⇡ �T

Hs ⇠ 1/d

TWO SPHERES AT DIFFERENT TEMPERATURES

FORCE BETWEEN TWO SPHERES (SIO2)• Dipole approximation, one reflection: assume radius

• Force on sphere 2: attraction (solid lines) and repulsion (dashed lines)

10-3

10-2

10-1

1

4 5 6 7 8 9 10 15 20

F [1

0-18 N

]

d [µm]

� d-2

T1=0 K,T2=0 KT1=300 K, T2=300 K

T1=0 K, T2=300 KT1=300 K, T2=0 K

10-3

10-2

10-1

1

4 5 6 7 8 9 10 15 20

F [1

0-18 N

]

d [µm]

� d-2

T1=300 K,T2=300 KT1=0 K, T2=0 K

T1=0 K, T2=300 KT1=300 K, T2=0 K

R⌧ d,�T =~c

KBT

Tenv = 0K Tenv = 300K

• Oscillations from due to interference of reflected and non-reflected radiation. Set by material resonances.

• Stable equilibrium positions.• Self-propelled pairs: equal acceleration in the same direction.

F↵↵

CASIMIR LEVITATION

• Non-equilibrium situation:

• Hot microsphere levitates above a cold dielectric plate

• If sphere cools down (including heat transfer) it will fall down

OUTLOOK / NEW DIRECTIONS Radiation/Transfer: effect of shape? e.g. non-parallel cylinders, disorder (roughness)?

Fluctuation of forces / radiation / transfer? distribution functions? Related to friction (Einstein relation): Quantum friction?

Relation to random matrices?

Dynamic effects: Radiation due to motion

...