Creation of Optical Cat and GKP States Using Shaped Free Electrons

Cat states and Gottesman-Kitaev-Preskill (GKP) states play a key role in quantum computation and communication with continuous variables. The creation of such states relies on strong nonlinear light-matter interactions, which are widely available in microwave frequencies as in circuit quantum electrodynamics platforms. However, strong nonlinearities are hard to come by in optical frequencies, severely limiting the development and applications of quantum-information science with continuous variable in the optical range. Here we propose using the strong interaction of free electrons with light to implement the desired nonlinear mechanism, showing its implication by creating optical cat and GKP states. The key to our finding is identifying conditions on the electron for which its interaction mimics the conditional displacement quantum gate. The strong interactions can be realized by phase matching of free electrons with photonic structures such as optical waveguides and photonic crystals in an ultrafast transmission electron microscope (UTEM). Our approach enables the generation of optical GKP states with above 10 dB squeezing and fidelities above 90% at postselection probability of 10%, even reaching >30% using an initially squeezed-vacuum state. We analyze the different factors that affect the fidelity, such as electron dispersion, inhomogeneity, nonideal interaction, and limited detection efficiency. Furthermore, the free-electron interaction allows two qubit gates between a pair of GKP states, which can entangle them into a GKP Bell state. We present a roadmap for realizing such experiments in a UTEM. Since electrons can interact resonantly with light across the electromagnetic spectrum, our approach could apply for a generation of cat and GKP states also in other platforms of free-electron radiation, from klystrons to free-electron lasers.