Escaping Heisenberg’s uncertainty principle

We discuss how one can escape Heisenberg's uncertainty principl and measure position and momentum in modular fashion simultaneously. In order to make use of this for sensing, one needs to prepare the mechanical system in a so-called grid state.

In 2000 Gottesman, Kitaev and Preskil formulated a proposal to encode a qubit in an oscillator (e.g. a cavity mode) using such grid states. We discuss several ways to prepare grid states in circuit-QED using a dispersive cavity-qubit coupling or by breeding from Schrödinger cat states. We finally show that preparing an oscillator in such state allows one to determine both parameters of a displacement with an accuracy which scales inversely with the square root of the number of photons in the oscillator state. For squeezed or coherent states this accuracy is a constant, independent of photon number.