Magnus Expansion and gaussian stochastic process?

[wintention]

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

 

[eljose79]

Hi wintention,

Thank you for your post and for sharing your calculations and findings. It seems like you are studying the effects of noise on a Spin-1/2 system controlled by an rf-pulse. Your results are interesting and it's great that you are looking into the connection between these results and known properties of the normal distributed variable n(t).

To answer your question, the connection between your results and the properties of the normal distributed variable can be explained using the concept of stochastic calculus. In particular, the properties of the normal distributed variable can be described by its mean, variance, and autocorrelation function.

The mean of a normal distributed variable is given by its expected value, which in your case is \overline{n}(t) = \lambda \neq 0. This means that, on average, the value of n(t) is equal to \lambda. This is reflected in your first-order term, where you have the factor of \lambda multiplying your integral.

The variance of a normal distributed variable is a measure of its spread or variability. It is given by the square of its standard deviation, which is denoted by \sigma_n(t). In your case, the variance is not explicitly included in your results, but it can be inferred from the fact that n(t) is a normal distributed variable.

Finally, the autocorrelation function of a normal distributed variable describes the correlation between its values at different times. In your case, this is reflected in your second-order term, where you have the integral of n(t_1)n(t_2). This term represents the correlation between the noise at two different times t_1 and t_2.

I hope this helps to clarify the connection between your results and the properties of the normal distributed variable. As for further reading, I would recommend looking into stochastic calculus and its applications in physics. There are many resources available online and in textbooks that can provide a more in-depth explanation of these concepts.

Best of luck with your research!