Quantum dynamics of two
Brownian particles
A. O. Caldeira
IFGW-UNICAMP
Outline
a) Introduction
b) Alternative model and effective coupling
c) Quantum dynamics
d) Results
e) Conclusions
Introduction
Equation of motion of a classical Brownian particle
mq   q  V q   f t  where
'
f t   0 and
f t  f t '  2k BT t  t '
a) The phenomenological
approach
Phenomenological
approach
Dissipative systems are such that
H = HS + Hint + HR
Two options:
i) simple model for HR + classical constraint
ii) realistic model for HR
Choosing the first one
pk2 1
p2
 mk k2 qk2 ;
HS 
 V q  ; H int   Ck qqk ; H R  
2
2m
k 2mk
k
Ck2 q 2
H CT  
2
k 2 mk k
Defining the spectral function
 Ck2
J    
   k 
k 2mk k
one shows that the condition for ohmic dissipation q  is
 if
J    
 0 if

 
Strategy: trace over the variables of R on the time evolution of the
density operator of the entire system S+R

Effective dynamics depends only on 
Other forms of the same model
Ck
qk 
q
2 k
mk k
p2
H
 V (q) 
2m
where
Ck2
k 
mk k4

2
and

k
Ck
pk 
pk
2
mk k
pk2  k k2

( qk  q ) 2
2 k
2
 J ( ) 

2

k
Ck2
 (  k ) 
mk k
3


 k k  (  k )   (   )
k
Manifestly translational invariant if V(q)=0!
Mechanical analogue
V(q)
Manifestly translational invariant if V(q)=0!
If we write the Lagrangian of the whole system as
~
L  LS  LR  LI (notice there is no counter – term! )
where
~
LI 

~
~
Ck qq k with Ck  Ck k1
k
and go over to the Hamiltonian formalism, we recover
the original model (with the appropriate counter – term) ,
after the canonical transformation
p  p, q  q, pk  mk k qk
pk
and qk 
mk k
Two free Brownian particles (classical)
Two independent particles immersed in a medium, if acted
by no external force obey
mq1  2m q1  f1 (t )
f1 (t )  0
and
and

;
2m
f1 (t ) f1 (t )  4m kT (t  t )
mq2  2m q 2  f 2 (t )
f 2 (t )  0
where

where
f 2 (t ) f 2 (t )  4m kT (t  t )
Two free Brownian particles (classical)
q1  q2
m
if q 
, u  q1  q2 , M  2m and  
2
2
f1 (t )  f 2 (t )
Mq  2M q  f CM (t ) where f CM (t ) 
;
2
f CM (t )  0 and f CM (t ) f CM (t )  4M kT (t  t )
 u(t )  2 u (t )  f R (t )
f R (t )  0
and
where f R (t )  f1 (t )  f 2 (t ) ;
f R (t ) f R (t )  4 kT (t  t )
Alternative model and effective coupling
Single particle
O.S.Duarte and AOC
Phys. Rev. Lett 97
250601 (2006)
Going over to the Hamiltonian formulation + canonical
transformation
Alternative model and effective coupling
Single particle
modelling
counter -term
Equation of
motion
Damping
kernel
Fluctuating
force
becomes a
constant
Alternative model and effective coupling
Single particle

K ( r, t )    d 2k 2 k  k
k
0
Assumption
Resulting equation
Im  k(0) ()

cos kr cos t
Alternative model and effective coupling
Two particles
next page
Alternative model and effective coupling
Two particles
modelling
For the center of mass and relative coordinates
x1  x2
q
and u  x1  x2
2
Alternative model and effective coupling
Two particles
Quantum dynamics
O.S.Duarte and AOC
To appear PRB 2009
Tracing the bath variables from the time evolution of the full
density operator one gets
for the reduced density operator of the system
Quantum dynamics
Results
Initial reduced
density operator
x  yi
New variables are defined
qi  i
and i  xi  yi
in terms of
2
1  2
q1  q2
and v  1   2
and u  q1  q2
and  
as r 
2
2
z is the squeeze parameter
reduced density
operator at any
time
Results
Characteristic
function
Covariance
matrix
Eigenvalues of
the PT density
matrix
Logarithmic
negativity
Results
  1.0, kT  104 , k0L  0,   10
Results
  1.0, kT  10, k0 L  0,   10
Results
z  0, kT  5, k0 L  0,   10
Results
  1.0, kT  104 , z  0,   10
Results
  1.0, kT  104 , z  0.3,   10
Conclusions
1) Generalization of the conventional model properly
describes the dynamics of two Brownian particles.
2) Novel possible effects: static and dynamical effective
interaction between the particles.
3) Possibility of two-particle bound states. Analogy with
other cases in condensed matter systems; Cooper
pairs, bipolarons etc.
4) Dynamical behaviour of entanglement for limiting
cases.