Parametric coupling between macroscopic quantum resonators

Abstract

Time-dependent linear coupling between macroscopic quantum resonator modes\ngenerates both a parametric amplification also known as a {}"squeezing\noperation" and a beam splitter operation, analogous to quantum optical systems.\nThese operations, when applied properly, can robustly generate entanglement and\nsqueezing for the quantum resonator modes. Here, we present such coupling\nschemes between a nanomechanical resonator and a superconducting electrical\nresonator using applied microwave voltages as well as between two\nsuperconducting lumped-element electrical resonators using a r.f.\nSQUID-mediated tunable coupler. By calculating the logarithmic negativity of\nthe partially transposed density matrix, we quantitatively study the\nentanglement generated at finite temperatures. We also show that\ncharacterization of the nanomechanical resonator state after the quantum\noperations can be achieved by detecting the electrical resonator only. Thus,\none of the electrical resonator modes can act as a probe to measure the\nentanglement of the coupled systems and the degree of squeezing for the other\nresonator mode.\n