Hybrid systems for the dissipative preparation of mechanical Schr\"odinger cats. (arXiv:1802.01306v1 [quant-ph])

We present a method to implement two-phonon interactions between mechanical
resonators and spin qubits in hybrid setups, and show that these systems can be
applied for the dissipative generation of nonclassical mechanical states. In
particular, we demonstrate that the implementation a two-phonon Jaynes-Cummings
Hamiltonian under coherent driving of the qubit yields a dissipative phase
transition with similarities to the one predicted in the model of the
degenerate parametric oscillator: beyond a certain threshold in the driving
amplitude, the driven-dissipative system sustains a steady state consisting of
a "jumping cat", i.e., a Schr\"odinger cat undergoing random jumps between two
phases. We consider realistic setups and show that, in samples within reach of
current technology, the low rate at which jumps between Schr\"odinger cat
states occur allows these to survive with fidelities $>0.99$ for longer than
one milisecond, without the need of any particular protocol or preparation.

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