m |
|||
| Line 7: | Line 7: | ||
\nu\theta(\eta-\mu)\sigma | \nu\theta(\eta-\mu)\sigma | ||
+\mu\eta(\nu-\theta){\sigma}^\dagger | +\mu\eta(\nu-\theta){\sigma}^\dagger | ||
| − | +[(1-\mu)\nu\eta(1-\theta) | + | +[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma). |
</m> | </m> | ||
</center> | </center> | ||
<m>\sigma</m> being the annihilation operator of a two-level system (one of the Pauli matrices), <m>\sigma^\dagger</m> its conjugate, <m>\mu,\nu,\eta,\theta</m> boolean variables (0 or 1), the following general commutator reads in normal order:
<m> [{\sigma}^{\dagger\mu}\sigma^\nu,{\sigma}^{\dagger\eta}\sigma^\theta] = \nu\theta(\eta-\mu)\sigma +\mu\eta(\nu-\theta){\sigma}^\dagger +[(1-\mu)\nu\eta(1-\theta)-\mu(1-\nu)(1-\eta)\theta](1-2{\sigma}^\dagger\sigma). </m>
I sometimes need this formula but always have to derive it again, which is very annoying (see this). (Unrelatedly, I also hope it is the last time I typeset something with texvc, as I plan to move to MathJax).