1. Quantum dynamics of two Brownian particles
A. O. Caldeira
IFGW-UNICAMP

2. Outline a) Introduction b) Alternative model and effective coupling
c) Quantum dynamics
d) Results
e) Conclusions

3. Introduction
Equation of motion of a classical Brownian particle

4. Phenomenological approach

6. Other forms of the same model
where
Manifestly translational invariant if V(q)=0!

7. Mechanical analogue
V(q)
Manifestly translational invariant if V(q)=0!

8. If we write the Lagrangian of the whole system as
(notice there is no counter – term! )
where
and go over to the Hamiltonian formalism, we recover the original model (with the appropriate counter – term) , after the canonical transformation

9. Two free Brownian particles (classical)
Two independent particles immersed in a medium, if acted by no external force obey

10. Two free Brownian particles (classical)

11. Alternative model and effective coupling
O.S.Duarte and AOC
Phys. Rev. Lett 97
(2006)
Single particle
Going over to the Hamiltonian formulation + canonical transformation

12. Alternative model and effective coupling
Single particle
modelling
becomes a constant
counter -term
Equation of motion
Damping kernel
Fluctuating force

13. Alternative model and effective coupling
Single particle
Assumption
Resulting equation

14. Alternative model and effective coupling
Two particles
next page

15. Alternative model and effective coupling
Two particles
modelling
For the center of mass and relative coordinates

16. Alternative model and effective coupling
Two particles

17. O.S.Duarte and AOC
To appear PRB 2009
Quantum dynamics
Tracing the bath variables from the time evolution of the full
density operator one gets
for the reduced density operator of the system

18. Quantum dynamics

19. Results Initial reduced density operator
New variables are defined in terms of as
and
z is the squeeze parameter
reduced density operator at any time

20. Results Characteristic function Covariance matrix
Eigenvalues of the PT density matrix
Logarithmic negativity

21. Results

22. Results

23. Results

24. Results

25. Results

26. Conclusions
Generalization of the conventional model properly describes the dynamics of two Brownian particles.
Novel possible effects: static and dynamical effective interaction between the particles.
Possibility of two-particle bound states. Analogy with other cases in condensed matter systems; Cooper pairs, bipolarons etc.
Dynamical behaviour of entanglement for limiting cases.