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Is there a simple Hamiltonian describing a Lorentzian pulse of a given frequency, pulse length and linewidth? I want to act a Lorentzian pulse on a Bloch vector and model the result for both resonant and non-resonant pulses.

JJH
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    You might want to be more explicit about what you're asking. Can you provide more details of what purpose you want this Hamiltonian for? – user35952 Apr 09 '18 at 12:18
  • Thanks, I want to be able to act a Lorentzian pulse onto any Bloch vector (on- and off-resonant) and see how it rotates. – JJH Apr 09 '18 at 12:25
  • Then it might be just an operator acting on the quantum state (here the Bloch vector) and not a Hamiltonian as such. – user35952 Apr 09 '18 at 12:28

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You can describe action of any pulse of magnetic field $B(t)$ on a Hamiltonian system with non-zero magnetic moment in this approximate way:

$$ H(t) = H_0 - \vec{\mu} \cdot \vec n \,B(t) $$

where $H_0$ is Hamiltonian describing the internal forces in the system, $\vec \mu$ is vector of magnetic moment of the system, $\vec{n}$ is direction of the magnetic field and $B(t)$ is magnetic field at the point where the system is, at time $t$.

A pulse that is Lorentzian function of time would be something like

$$ B(t) = B_0 \frac{\tau^2}{(t-t_0)^2 + \tau^2} $$ where $B_0$ is the highest magnetic induction that pulse gives, $t_0$ is time where the pulse is most intense and $\tau$ characterizes duration of the pulse (such pulse is infinitely long, but $\tau$ gives time interval during which the pulse is "strong").

Ján Lalinský
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  • Thanks for your answer! When you define $B_0$ as the highest magnetic induction, is this equivalent to the maximum amplitude in the time domain? E.g. if I've got a pulse that is Lorentzian in frequency centred at 97.5MHz with a pulse width of 5MHz, I believe it will have the equation $\frac{2.5}{\pi((x - 97.5)^2 + 2.5^2)}$ in the frequency domain. In this case, how would the equivalent of $B_0$ be defined? – JJH Apr 11 '18 at 10:07
  • Well, a pulse that is Lorentzian in frequency won't be Lorentzian in time. So if you have the former, my answer does not apply. The time dependence of the pulse would be that of exponentially decaying oscillating function of time. See for example http://mathworld.wolfram.com/FourierTransformLorentzianFunction.html – Ján Lalinský Apr 11 '18 at 20:28
  • So is it correct that I can define a time dependent Hamiltonian of a pulse on say the $z$ axis as $e^{−2πix_0t−Γπ|t|}\sigma_z$? And is there then a way of moving this into a rotating frame so that the evolution of a state due to this pulse can be described by some unitary operator $U=e^{-iHt}$? – JJH Apr 12 '18 at 09:17