Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.
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Transcript of Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.
Quantum dynamics of two Brownian particles
A. O. CaldeiraIFGW-UNICAMP
Outline
a) Introduction
b) Alternative model and effective coupling
c) Quantum dynamics
d) Results
e) Conclusions
Introduction
tfqVqqm ' where
0tf and
'2' ttTktftf B
Equation of motion of a classical Brownian particle
qVm
pH S
2
2
; kk
kqqCH int ; 222
2
1
2 kkkk k
kR qm
m
pH ;
k kk
kCT m
qCH 2
22
2
Phenomenological approach
Defining the spectral function
kk kk
k
m
CJ
2
2
one shows that the condition for ohmic dissipation q is
if
ifJ
0
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
k
kk
kkk
kk
kk p
mC
pqm
Cq
22and
2222
)(22
)(2
qqp
qVm
pH k
kk
k k
k
)()(2
)(2
)(
3
2
4
2
kk
kk
kk kk
k
kk
kk m
CJ
mC
where
Manifestly translational invariant if V(q)=0!
V(q)
Mechanical analogue
Manifestly translational invariant if V(q)=0!
If we write the Lagrangian of the whole system as
IRS LLLL ~
where 1~with
~~ kkkkk
kI CCqqCL
and go over to the Hamiltonian formalism, we recover the original model (with the appropriate counter – term) , after the canonical transformation
(notice there is no counter – term! )
kk
kkkkkk m
pqqmpqqpp
and,,
Two free Brownian particles (classical)
)(4)()(and0)(
;2
where)(2
111
111
ttkTmtftftfm
tfqmqm
)(4)()(and0)(
where)(2
222
222
ttkTmtftftf
tfqmqm
Two independent particles immersed in a medium, if acted by no external force obey
Two free Brownian particles (classical)
2and2,,
2if 21
21 mmMqqu
qqq
)(4)()(and0)(
;2
)()()(where)(2 21
ttkTMtftftf
tftftftfqMqM
CMCMCM
CMCM
)(4)()(and0)(
;)()()(where)()(2)( 21
ttkTtftftf
tftftftftutu
RRR
RR
Alternative model and effective coupling
Single particle
Going over to the Hamiltonian formulation + canonical transformation
O.S.Duarte and AOCPhys. Rev. Lett 97250601 (2006)
Alternative model and effective coupling
Single particle
modelling
counter -term becomes a constant
Equation of motion
Damping kernel
Fluctuating force
Alternative model and effective coupling
Single particle(0)
2
0
Im ( )( , ) 2 cos cosk
k kk
K r t d k kr t
Assumption
Resulting equation
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
1 21 2and
2
x xq u x x
Alternative model and effective coupling
Two particles
Quantum dynamics O.S.Duarte and AOCTo appear PRB 2009
Tracing the bath variables from the time evolution of the fulldensity operator one gets
for the reduced density operator of the system
Quantum dynamics
Results
Initial reduced density operator
and2
i ii i i i
x yq x y
1 21 2and
2v
1 2
1 2and2
q qr u q q
New variables are defined in terms of
as and
reduced density operator at any time
z is the squeeze parameter
Results
Characteristic function
Covariance matrix
Eigenvalues of the PT density matrix
Logarithmic negativity
Results
401.0, 10 , 0, 10kT k L
Results
01.0, 10, 0, 10kT k L
Results
00, 5, 0, 10z kT k L
Results
41.0, 10 , 0, 10kT z
Results
41.0, 10 , 0.3, 10kT z
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.