Resonant Enhancement and Dissipation in Nonequilibrium

van der Waals Forces

Adam E. Cohen (Stanford)Shaul Mukamel (UC Irvine)

Intermolecular forces and optical response

van der Waals: RTbVVap

)(2

a ~ index of refraction (empirically)

McLachlan:

0

2 )()()(2

)(

diirJrU ba

0

2 )()()()(n

nbnaB iirTJkrF

T = 0:

T > 0: ‘ =polarizability

J = coupling

/2 TnkBn

Gauss: if 23

13

21

1~ rdrdrr

f n

and f is insensible between macroscopic objects, then 6n

3 + 3 = 6

Relating Nonlinear Optical Response to Intermolecular ForcesStart with perturbations to the individual molecules:

)()( 0 tEpHtH aaaa )()( 0 tEpHtH bbbb

121a2a

t t t

12)2(

a11a1)1(

a0aa dtdt)t(E)t(E)t,t,t(Rdt)t(E)t,t(Rp)t(p

Can calculate or measure R(1), R(2), … for any initial state.

Response given by Volterra series:

To calculate R(n): L

i

)t()1()0( LLL Liouville superoperator

01n

n

1n)n( )t(p̂)t(p̂)t(p̂i

!n1)t,,t,t(R

T

Time-ordering superoperator X,AXA

Commutator

X,AXA 21

Anticommutator

titi

AeetA 00)(ˆ LL

Liou. Space interaction picture

)A(A 00Tr

Expectation value

coordinate operator classical source

Do the nonlinear response functions completely describe a molecule?

01n

n

1n )t(p̂)t(p̂)t(p̂i!n

1)t,,t,t(Rn

T terms

In a quantum system or a classical ensemble, fluctuations have a life of their own

Calculate response of fluctuations to a perturbation:

01nma

n

1nma )t(p̂)t(p̂)t(p̂)t(p̂i!n

1)t,,t,t,,t(Rnm

T

terms terms

R++ and R+- are related by the Fluctuation-Dissipation Theorem (FDT)

Call R+...+-...- Generalized Response Functions (GRFs)

Causality Generalized K-K relations

Thermal equilibrium Generalized FDT

(Compare with )

Two coupled moleculesCoupled molecules:

baba pptJHHtH )()( 00

1212

t t t

12)2(

111)1(

0baba dtdt)t(J)t(J)t,t,t(R~dt)t(J)t,t(R~pp)t(p)t(p

Want to evaluate:

L

i

)t()1()0(b

)0(a LLLL

Liouville superoperatorAgain

01b1anbnaba

n

1n)n( )t(p̂)t(p̂)t(p̂)t(p̂)t(p̂)t(p̂i

!n1)t,,t,t(R~

TUsing superoperator algebra, we can factor the joint response function:

n

mnbnan

n tttRtttRtttR mn mmn - m

0111

)( ),,,(),,,(),,,(~ terms terms terms terms

The joint response of the coupled molecules depends on all GRFsof the individual molecules.

Joint response function:

Example: Coupled Harmonic Oscillators

Phys. Rev. Lett. 91, 233202 (2003)

),(),(),(),(),(~11111

)1( ttRttRttRttRttR baba

1st order response to coupling J(t)papb:

Frequency domain: )()()1( Jpp ba

Steady state coupling: 2)1( )0(21 JF

Reproduces McLachlan formula for Ta = Tb

Time domain: 111)1( )(),(~ dttJttRpp

t

ba

0.51

1.52

0.5

1

1.5

2

- 20

- 10

0

10

0.51

1.52

0.5

1

1.5

2

b/a

b/a

(1)(0)

Dissipation between coupled SHOs

For time-varying J, need: )('')(')()1( i

Force Dissipation

Possibility of negative friction

-20

2

0.5

1

1.5

2

-10

0

10

-20

2

0.5

1

1.5

2

b/a

Re[(1)] Im[(1)]

-20

2

0.5

1

1.5

2

-5

0

5

-20

2

0.5

1

1.5

2

b/a

Example: FRET forceFluorescence Resonance Energy Transfer (FRET) is mediated by thesame dipole-dipole interaction that mediates the vdW force.

Do not fret for it leads only to evil.--Psalm 37

J. Phys. Chem. A 107 (19) 3633 (2003)

Forster rate of FRET:

d

nf

rck ad

dFRET

0

35

''

60

23

)()()(

83

orientational factordonor emission spectrum

lifetime of donor

Interaction energy from FRET:

d

nf

rcU ad

dFRET

0

35

'

60

23

)()()(

163

Kramers-Kronig relation between kFRET and UFRET

UFRET can also be thought of as optical trappingof acceptor in near-field of excited donor.

Sample calculationChlorophyll b in diethyl ether

-20

0

20

40

60

80

100

550 600 650 700 750 (nm)

Im(a) fD

-20

0

20

40

60

80

100

550 600 650 700 750

(nm)

Re(a) fD

FRET FRET force

FRET force may be either attractive or repulsive

FRET force may be much stronger than vdW force

Possibilities for experimental verification

• NLO effects in critical systems (gasses, binary mixtures, polymers)

• Conformational changes in tethered bichromophores• Concentration quenching• Solid state measurements (Casimir-type)

Conclusions

• Quantum ensemble described by Generalized Response Functions (GRFs)

• Response functions of two coupled systems may be expressed in terms of the GRFs of the constituents

• vdW forces between objects at different temperatures or in relative motion show resonant enhancement and (possibly negative) dissipation

• A mechanical force accompanies FRET

Acknowledgments

Professor Shaul Mukamel (UCI)

$$ Hertz Foundation $$