Calculation of Multi-Time correlation function

[Gernot Pfanner]

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).
*Well*, as stated before, I am now looking -really badly indeed- for a
similar reasoning, if there are four times involved.
In this spirit
With many thanks for your efforts
Yours Gernot

[Arnold Neumaier]

Gernot Pfanner schrieb:

> 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).
> *Well*, as stated before, I am now looking -really badly indeed- for a
> similar reasoning, if there are four times involved.

What do you need the correlations for?

In Gardiner's book on quantum noise he shows how to get multi-time
correlations in which the times have a palindromic pattern
t_1 t_2 ... t_n t_{n-1} ... t_1, where t_1 <= t_2 <= ... <= t_n.
By setting some operators to the identity, one can get non-palindromic
correlations. But only if the time ordering is monotone or unimodal
(first up then down). I think only such correlations are needed in
applications.


Arnold Neumaier