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

Phenomenological approach

Other forms of the same model where 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 (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

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

Two free Brownian particles (classical)

Alternative model and effective coupling O.S.Duarte and AOC Phys. Rev. Lett 97 250601 (2006) Single particle Going over to the Hamiltonian formulation + canonical transformation

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

Alternative model and effective coupling Single particle 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

Alternative model and effective coupling Two particles

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

Quantum dynamics

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

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

Results

Results

Results

Results

Results

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.