Entanglement Thresholds of Doubly Parametric Quantum Transducers

Doubly parametric quantum transducers, such as electro-optomechanical devices, show promise for providing the critical link between quantum information encoded in highly disparate frequencies such as in the optical and microwave domains. This technology would enable long-distance networking of superconducting quantum computers. Rapid experimental progress has resulted in impressive reductions in decoherence from mechanisms such as thermal noise, loss, and limited cooperativities. However, the fundamental requirements on transducer parameters necessary to achieve quantum operation have yet to be characterized. In this work we find simple, protocol-independent expressions for the necessary and sufficient conditions under which doubly parametric transducers in the resolved-sideband, steady-state limit are capable of entangling optical and microwave modes. Our analysis treats the transducer as a two-mode bosonic Gaussian channel capable of both beamsplitter-type and two-mode squeezing-type interactions between optical and microwave modes. For the beamsplitter-type interaction, we find parameter thresholds that distinguish regions of the channel's separability, capacity for bound entanglement, and capacity for distillable entanglement. By contrast, the two-mode squeezing-type interaction always produces distillable entanglement with no restrictions on temperature, cooperativities, or losses. Counterintuitively, for both interactions, we find that achieving quantum operation does not require either a quantum cooperativity exceeding one, or ground-state cooling of the mediating mode. Finally, we discuss where two state-of-the-art implementations are relative to these thresholds and show that current devices operating in either mode of operation are in principle capable of entangling optical and microwave modes.