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Markovian Master Equation and Uncorrelated States

  • Thread starter Kreizhn
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1. Homework Statement
In deriving the Markovian master equation for a weakly coupled system+environment scensario, we have
[tex] \frac{\partial}{\partial t} P \rho(t) = \alpha^2 \int_{t_0}^t P L(t) L(s) P \rho(s) ds [/tex]
where [itex] \alpha [/itex] is the coupling strenght, [itex] P\rho = \left( \mathrm{Tr} \rho \right) \otimes \rho_B [/itex] for [itex] \rho_B [/itex] the initial environment state, [itex] Q\rho = (\mathcal I - P ) \rho [/itex] for [itex] \mathcal I [/itex] the identity superoperator, and L the Liouvillian of the system satisfying [itex] L(t)\rho = -i[H_I(t), \rho(t)] [/itex] where H is the interaction Hamiltonian. My goal is to show that with the knowledge, if we have an initially uncorrelated state
[tex] \rho(t_0) = \rho_A(t_0) \otimes \rho_B(t_0) [/tex]
then the state remains uncorrelated for all time [itex] t \geq 0 [/itex]

3. The Attempt at a Solution
This pre-master equation comes from the fact that we've assumed the propagator [itex] G(s,t) = \mathrm{exp}\left[ \alpha \int_s^t QL(\tau) d\tau \right] [/itex] is approximated by identity (which occurs in the weak coupling limit). I need to show that in this weak coupling limit, the states do not become entangled (intuitively reasonable), though I can't think of how to show that the evolution according to the above differo-integral equation keeps the states uncorrelated.
 
743
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Nobody has any ideas?
 

gabbagabbahey

Homework Helper
Gold Member
5,001
6
Well, this is just an idea but; you might try showing that [tex]\rho(t) = \rho_A(t) \otimes \rho_B(t) [/tex] is a solution, and then perhaps see if there might be some uniqueness theorem you could appeal to that guarantees it's the only solution given the initial condition [tex]\rho(t_0) = \rho_A(t_0) \otimes \rho_B(t_0)[/tex]
 

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