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Magnus Expansion and gaussian stochastic process?

  1. Oct 2, 2011 #1
    Hi,

    I do some calculations on a rf-pulse controlled Spin-1/2 system influenced by noise given by a normal distributed random variable [itex]n(t)[/itex] (which is, I guess, a gaussian stochastic process, as [itex]n(t)[/itex] is a gaussian distributed random variable for all [itex]t[/itex]).

    Using the Magnus-Expansion
    [itex]
    \exp\left[-i\left(\mu^{\left(1\right)}-\frac{i}{2}\mu^{\left(2\right)}+\dots\right)\right]
    [/itex]
    on the time-ordered time evolution operator, I found (in the case of a control pulse in [itex]y[/itex]-direction and noise coupling in [itex]z[/itex]-direction, for example) the first two terms to be something like
    [itex]
    \mu^{\left(1\right)}= \int\limits_{0}^{\tau_{p}} f \left( t \right) n \left( t \right) \mathrm{dt}\cdot\sigma_z + \int\limits_{0}^{\tau_{p}} g \left( t \right) n \left( t \right) \mathrm{dt}\cdot\sigma_x
    [/itex]
    and
    [itex]
    \mu^{\left(2\right)}= \int \limits_{0}^{\tau_{p}}\int\limits_{0}^{t_1} h \left(t_{1},t_{2}\right) n\left(t_{1}\right)n\left(t_{2}\right)\mathrm{dt_{2}}\mathrm{dt_{1}}\cdot\sigma_y
    [/itex],
    where [itex]f(t)[/itex], [itex]g(t)[/itex] and [itex]h(t_1,t_2)[/itex] are trigonometric functions.

    Now, I wonder how to connect these results (and those for terms of higher orders) with known properties of the normal distributed variable [itex]n(t)[/itex], like the (auto-)correlation function and [itex]\overline{n}(t) = \lambda \neq 0[/itex].

    I did some research and found
    [itex]
    \mu^{\left(1\right)} = \int\limits_{0}^{\tau_{p}} f \left( t \right) n \left( t \right) \mathrm{dt}\cdot\sigma_z + \int\limits_{0}^{\tau_{p}} g \left( t \right) n \left( t \right) \mathrm{dt}\cdot\sigma_x
    [/itex]
    [itex]
    = \lambda\int\limits_{0}^{\tau_{p}} f \left( t \right) \mathrm{dt}\cdot\sigma_z + \lambda\int\limits_{0}^{\tau_{p}} g \left( t \right) \mathrm{dt}\cdot\sigma_x
    [/itex].
    However, I'm not quite convinced here, as I didn't find a source explaining this mathematically to me, yet.

    So, could anybody help me with that? I'm relatively new to those stochastic problems, so may be an advice where to read about them in a physical context could be helpful, too.

    Thank you very much
    -wintention
     
  2. jcsd
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