Directional amplifiers are an important resource in quantum information processing, as they protect sensitive quantum systems from excess noise. Here, we propose an implementation ofphase-preserving and phase-sensitive directional amplifiers for microwave signals in an electromechanical setup comprising two microwave cavities and two mechanical resonators. We show that both can reach their respective quantum limits on added noise. In the reverse direction, they emit thermal noise stemming from the mechanical resonators and we discuss how this noise can be suppressed, a crucial aspect for technological applications. The isolation bandwidth in both is of the order of the mechanical linewidth divided by the amplitude gain. We derive the bandwidth and gain-bandwidth product for both and find that the phase-sensitive amplifier has an unlimited gain-bandwidth product. Our study represents an important step toward flexible, on-chip integrated nonreciprocal amplifiers of microwave signals.
Classically, the tendency towards spontaneous synchronization is strongest if the natural frequencies of the self-oscillators are as close as possible. We show that this wisdom failsin the deep quantum regime, where the uncertainty of amplitude narrows down to the level of single quanta. Under these circumstances identical self-oscillators cannot synchronize and detuning their frequencies can actually help synchronization. The effect can be understood in a simple picture: Interaction requires an exchange of energy. In the quantum regime, the possible quanta of energy are discrete. If the extractable energy of one oscillator does not exactly match the amount the second oscillator may absorb, interaction, and thereby synchronization is blocked. We demon- strate this effect, which we coin quantum synchronization blockade, in the minimal example of two Kerr-type self-oscillators and predict consequences for small oscillator networks, where synchronization between blocked oscillators can be mediated via a detuned oscillator. We also propose concrete implementations with super- conducting circuits and trapped ions. This paves the way for investigations of new quantum synchronization phenomena in oscillator networks both theoretically and experimentally.