All evidence so far suggests that the absolute spatial orientation of an experiment never affects its outcome. This is reflected in the Standard Model of physics by requiring all particles and fields to be invariant under Lorentz transformations. The
most well-known test of this important cornerstone of physics are Michelson-Morley-type experimentscite{MM, Herrmann2009,Eisele2009} verifying the isotropy of the speed of light. Lorentz symmetry also implies that the kinetic energy of an electron should be independent of the direction of its velocity, textit{i.e.,} its dispersion relation should be isotropic in space. In this work, we search for violation of Lorentz symmetry for electrons by performing an electronic analogue of a Michelson-Morley experiment. We split an electron-wavepacket bound inside a calcium ion into two parts with different orientations and recombine them after a time evolution of 95ms. As the Earth rotates, the absolute spatial orientation of the wavepackets changes and anisotropies in the electron dispersion would modify the phase of the interference signal. To remove noise, we prepare a pair of ions in a decoherence-free subspace, thereby rejecting magnetic field fluctuations common to both ionscite{Roos2006}. After a 23 hour measurement, we limit the energy variations to $htimes 11$ mHz ($h$ is Plancks constant), verifying that Lorentz symmetry is preserved at the level of $1times10^{-18}$. We improve on the Lorentz-violation limits for the electron by two orders of magnitudecite{Hohensee2013c}. We can also interpret our result as testing the rotational invariance of the Coloumb potential, improving limits on rotational anisotropies in the speed of light by a factor of fivecite{Herrmann2009,Eisele2009}. Our experiment demonstrates the potential of quantum information techniques in the search for physics beyond the Standard Model.
