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From: OsherD on 27 Apr 2010 02:37
From Osher Doctorow

Wei Cui, Zai-Rong Xi, and Yu Pan of Institute of Systems Science
Beijing China, in "Non-Markovian entanglement dynamics in coupled
superconducting qubit systems," arXiv 1004.4303 v1 [quant-ph] 24 Apr
2010, develop a Non-Markovian Master Equation and with analysis and
simulation of Josephson junctions they find that entanglement can be
open-loop controlled by adjusting temperature, the ratio r of
reservoir cutoff frequency to system oscillator frequency, and
superconducting phases PHI_k in superconducting qubit systems. The
whole scenario is one of Memory including strong Memory effects.

I will try to discuss the specific equations later, but readers should
recall that Memory is key to Probable Causation/Influence (PI) as
discussed in this thread.

Osher Doctorow
From: OsherD on 27 Apr 2010 03:12
From Osher Doctorow

The authors compare the Markovian and the derived Non-Markovian Master
Equations, with the latter rather than the former being the applicable
one.

The Master Equations have the left hand side:

1) Dt(rho(t)), where rho(t) is the density matrix.

The right hand side involve two terms (I will refer to the Non-
Markovian case here), the first being:

2) -i[Hs, rho] where Hs(t) is the Hamiltonian and describes the
coherent part of the time evolution.

The second term is a summation that involves:

3) D[L]rho = LrhoL^(+) - (1/2)L^(+)Lrho - (1/2)rhoL^(+)L with +
representing here the Hermitian conjugate dagger symbol.

The super-operator D[L] in the Non-Markovian Master Equation and
evolution equation is applied to C_m or Cm for short rather than
arbitrary L, where Cm are time dependent system operators through
various decay channels labelled by m with corresponding time-dependent
decay rates DELTA_m(t). If > = 1 DELTA_m(t) is negative even
temporarily, then there are strong Memory effects.

In addition to super-operator D[L], the second term involves:

4) DELTA(t) and gamma(t), respectively diffusive and damping terms
which are integrals from 0 to t of respectively noise and dissipation
kernels u(tau) and k(tau) d(tau) with also a cosine or a sine factor
in the integrand (cos(w tau), sin(w tau)). The u(tau) and k(tau)
factors in turn have integral representations from 0 to infinity with
respect to dw.

Osher Doctorow
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