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- 1. Measurement-induced long-distance entanglement of superconducting qubits using optomechanical transducers Ondřej Černotík and Klemens Hammerer Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Germany - Circuit QED Optomechanical transduction Entanglement by measurement Adiabatic elimination Force sensing Results Max Planck Institute for Gravitaitonal Physics (Albert Einstein Institute) Acknowledgements We thank Pertti Hakonen for useful discussions. This work was funded by the European Commission (FP7-Programme) through iQUOEMS (Grant Agreement No. 323924). We acknowledge support by DFG through QUEST and by the cluster system team at the Leibniz University Hannover. Email: Ondrej.Cernotik@itp.uni-hannover.de References [1] R. Andrews, et al., Nature Physics 10, 321 (2014). [2] T. Bagci, et al., Nature 507, 81 (2014). [3] K. Stannigel et al., PRL 105, 220501 (2010). [4] N. Roch et al., PRL 112, 170501 (2014). [5] O. Černotík, D. V. Vasilyev, and K. Hammerer, PRA 92, 012124 (2015). [6] O. Černotík and K. Hammerer, arXiv:1512.00768. Superconducting structures offer a promising platform for quantum computation. Due to transition frequencies in the microwave range, up-conversion to optical frequencies is needed to enable long-distance signal transmission. Mechanical oscillators couple to both microwave and optical fields. They are thus suitable for converting signal from one spectral range to the other [1, 2] and can be used to connect superconducting systems via a room- temperature environment [3]. With the coupling , the qubit exerts a force on the mechanical oscillator. This force can be measured with an optomechanical system; the qubit states can be distinguished with measurement time At the same time, the mechanical bath disturbs the qubit, leading to its dephasing at a rate The measurement needs be faster than this dephasing which is achieved for strong optomechanical cooperativity, The system is too complicated to be treated analytically or numerically, so we adiabatically eliminate the transducer dynamics. We use the fact that the dynamics is Gaussian and can be described using the first and second statistical moments of its canonical operators. 0 1 2 3 4 5 Time (µs) 0.0 0.4 0.8 Concurrence Psucc=0.1 Psucc=0.5 0.0 0.2 0.4 0.6 0.8 1.0 Optical transmission τ 0.0 0.2 0.4 0.6 0.8 Concurrence η=1 η=0.6 η=0.2 0.0 0.2 0.4 0.6 0.8 1.0 Optical transmission τ Psucc=0.1 Psucc=0.5 Dispersive interaction of light with a qubit results in a phase shift on the field. After a sequential interaction with two qubits, the measurement of phase reveals the total spin, . Starting from the initial two-qubit state with , the entangled state can be prepared conditionally, as demonstrated by Roch et al. [4]. Equation of motion Formally, the system can be described using the stochastic master equation with the Hamiltonian describes decoherence of the qubits, is a Lindblad term, and accounts for the effect of the measurement. The scheme requires only moderately large optomechanical cooperativity and can be implemented with present technology. Remarkably, the protocol tolerates significant amount of optical loss. See [6] for more details and other experimental implementations. Effective dynamics After adiabatically eliminating the transducer degrees of freedom, the effective equation of motion is with Optical loss can also be included in the model. Imperfect transmission between the systems introduces additional dephasing of the first qubit while losses after the second system decrease the detection efficiency , see [6] for details. The covariance matrix of the conditional transducer state then obeys a deterministic Riccati equation which can be solved efficiently [5].