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Additional Formulae

These formulas are often computed using helper functions as part of the Hamiltonian construction process. When this is the case, the name and location of the helper function is specified.

\[ \Omega = \frac{\pi}{\tau_\pi} \]
  • \(\Omega \equiv\) Rabi frequency
  • \(\tau_{\pi} \equiv\) \(\pi\)-time
\[ I = \frac{\epsilon_0 c }{2}\left(\frac{\hbar\Omega}{e M_{12}}\right)^2 \]
  • \(M_{12} \equiv \langle 1| \hat{M}|2 \rangle \equiv\) Multipole transition matrix element

Used in intensity_from_laser and rabi_from_intensity

\[ E = \sqrt{\frac{2I}{\epsilon_0 c}} \]
  • \(I \equiv\) Laser intensity
  • \(E \equiv\) magnitude of laser E-field \(\vec{E}\)

Used in rabi_from_intensity

\[ \eta = \vec{k}\cdot\hat{b}\sqrt{\frac{\hbar}{2m\nu}} \]
  • \(\eta \equiv\) Lamb-Dicke parameter
  • \(\vec{k} \equiv\) Laser wavevector
  • \(\hat{b} \equiv\) Motional mode eigenvector
  • \(m \equiv\) Ion mass
  • \(\nu \equiv\) Motional mode eigenfrequency

Used in ConstructHamiltonian