Hi!
I want to calculate a second-order-correlation-function with four
different times, i.e. G^2=Trace[rho.A*(t1)B*(t2)B(t3)A(t4)] (where
*...adjoint operator, rho...density matrix in the Schrödinger picture).
Unfortunately, I do not know how to do that.
The problem is that the relationship to the corresponding Lindblad
equations is somewhat more difficult than for just two times. In the
ladder situation, it is quite obvious what to do. Firstly, I insert the
corresponding time evolution operators (system+bath) into the correlation
function
G(2)[t1,t2]=
Tr[rho.U*(t1).A*.U(t1).U*(t2).B*.U(t2).U*(t2).B.U(t2).U*(t1).A.U(t1)].
By cyclic permutation one may rewrite this to
G(2)[t1,t2]=Tr[U(tau).A.U(t1).rho.U*(t1)A*.U*(tau).B*.B] (where
tau=t2-t1). In the Markov approximation, I may trace over the bath
degrees, yielding G(2)[t1,t2]=Tr_{sys}[rho_[red}.B*.B]. Now I just have to
solve the appropriate Lindblad equation (for rho_{red} in depdendence of
tau) to get G(2) (see also e.g. [1])
*Well*, as stated before, I am now looking -really badly indeed- for the
extension to the four-times case.
In this spirit
With many thanks for your efforts
Yours Gernot
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[1] p25-27 in
http://deposit.d-nb.de/cgi-bin/dokserv?idn=977864693
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