Resonant Enhancement and
Dissipation in Nonequilibrium
van der Waals Forces
Adam E. Cohen (Stanford)
Shaul Mukamel (UC Irvine)
Intermolecular forces and optical response
Gauss: if f ~ 
3+3=6
1
r1  r2
n
d 3 r1d 3 r2
and f is insensible between macroscopic
objects, then n  6
a 

p

(V  b)  RT
van der Waals: 
2 
 V 
a ~ index of refraction (empirically)
McLachlan:


2
T = 0: U (r )   J (r ) 0  a (i ) b (i )d
2
T > 0: F (r )  k BTJ (r )

2
‘ (i ) (i )
 a n b n
n 0
J = coupling
 = polarizability
 n  2nk BT / 
Relating Nonlinear Optical Response to Intermolecular Forces
Start with perturbations to the individual molecules:
H b (t )  H b0  pb Eb (t )
H a (t )  H a0  pa Ea (t )
coordinate operator
classical source
Response given by Volterra series:
t
pa ( t )  pa
  R ( t ,t1 )Ea ( t1 )dt1 
0
(1)
a

t

t
(2)
R
a
 ( t ,t2 ,t1 )Ea ( t2 )Ea ( t1 )dt2 dt1  
  
Can calculate or measure R(1), R(2), … for any initial state.
To calculate R(n):
i

   L 
Liouville superoperator
L  L( 0 )  L( 1 ) ( t )
n
R
(n)
1i
( t ,t n , ,t1 )    T p̂ ( t ) p̂ ( t n ) p̂ ( t1 )
n!   
0
Commutator
Time-ordering superoperator
A X  A, X 
Anticommutator
A X 
1
2
A, X 
Liou. Space interaction picture
Aˆ (t )  e
i
L 0t

Ae
i
 L 0t

Expectation value
A 0  Tr ( A0 )
Do the nonlinear response functions
completely describe a molecule?
In a quantum system or a classical ensemble,
fluctuations have a life of their own
Calculate response of fluctuations to a perturbation:
n
m terms n terms


 
R   ( t a , ,t m ,t n , ,t1 ) 
(Compare with
n
terms
R
 
1i
  T p̂ ( t a ) p̂ ( t m ) p̂ ( t n ) p̂ ( t1 )
n!   
n
1i
( t ,t n , ,t1 )    T p̂ ( t ) p̂ ( t n ) p̂ ( t1 )
n!   
0
)
Call R+...+-...- Generalized Response Functions (GRFs)
R++ and R+- are related by the Fluctuation-Dissipation Theorem (FDT)
Causality  Generalized K-K relations
Thermal equilibrium  Generalized FDT
0
Two coupled molecules
Coupled molecules:
H (t )  H a0  H b0  J (t ) pa pb
Want to evaluate:
t
~
pa ( t ) pb ( t )  pa pb 0   R ( 1 ) ( t ,t1 )J ( t1 )dt1 

i

Again    L 
t

t
~( 2 )
R
 ( t ,t2 ,t1 )J ( t2 )J ( t1 )dt2 dt1  
  
Liouville superoperator
L  L (a0 )  L (b0 )  L ( 1 ) ( t )
Joint response function:
n
1i
~
R ( n ) ( t ,t n , ,t1 )    T  p̂a ( t ) p̂b ( t )  p̂a ( t n ) p̂b ( t n )  p̂a ( t1 ) p̂b ( t1 )
n!   
0
Using superoperator algebra, we can factor the joint response function:
n
~ (n)
R (t , tn ,, t1 )   Ra
m 0
n - m terms m terms


 


(t , t1 ,, t n ) Rb
m terms n  m terms


 


(t , tn ,, t1 )
The joint response of the coupled molecules depends on all GRFs
of the individual molecules.
Example: Coupled Harmonic Oscillators
1st order response to coupling J(t)papb:
~
R (1) (t , t1 )  Ra (t , t1 ) Rb (t , t1 )  Ra (t , t1 ) Rb (t , t1 )
t
Time domain:
pa pb 
~ (1)
R
 (t , t1 ) J (t1 )dt1

(1)
Frequency domain: pa pb   ( ) J ( )
1
2
Steady state coupling: F   (1) (0) J 2
Reproduces McLachlan formula for Ta = Tb
2
bb/ ba
1.5
1
(1)(0)
0.5
10
0
-
10
-
20
0.5
1
b/ a
1.5
2
Phys. Rev. Lett. 91, 233202 (2003)
Dissipation between coupled SHOs
(1)
For time-varying J, need:  ( )   ' ( )  i ' ' ( )
Force
Dissipation
2
2
1.5
bb/ ba
bb/ ba
1
1
0.5
10
Im[(1)]
Re[(1)]
0.5
1.5
0
-10
5
0
-5
-2
-2
0

2
Possibility of negative friction

0
2
Example: FRET force
Fluorescence Resonance Energy Transfer (FRET) is mediated by the
same dipole-dipole interaction that mediates the vdW force.
Do not fret for it leads only to evil.
--Psalm 37
Forster rate of FRET: k FRET
orientational factor
donor emission spectrum
3c 3 2

80 d r 6


0
f d ( ) a'' ( )
d
5
3
n ( )
lifetime of donor
Interaction energy from FRET: U FRET
3c 3 2

160 d r 6


0
f d ( ) a' ( )
d
5
3
n ( )
Kramers-Kronig relation between kFRET and UFRET
UFRET can also be thought of as optical trapping
of acceptor in near-field of excited donor.
J. Phys. Chem. A 107 (19) 3633 (2003)
Sample calculation
Chlorophyll b in diethyl ether
FRET force
FRET
100
100
80
80
Im(a)
Re(a)
fD
60
60
40
40
20
20
0
0
-20
550
600
650
 (nm )
700
750
-20
550
600
650
 (nm )
FRET force may be either attractive or repulsive
FRET force may be much stronger than vdW force
700
fD
750
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)
$$ Hertz Foundation $$