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Although superconducting systems provide a promising platform for quantum computing, their networking poses a challenge as they cannot be interfaced to light—the medium used to send quantum signals through channels at room temperature. We show that mechanical oscillators can mediate such coupling and light can be used to measure the joint state of two distant qubits. The measurement provides information on the total spin of the two qubits such that entangled qubit states can be postselected. Our scheme works in analogy to experimental technique already established in the microwave domain but employs an optical channel at room temperature. The use of light greatly enhances the distance over which the qubits can become entangled. The generalization to the optical domain—although relatively straightforward from the experimental point of view—is highly nontrivial and requires a systematic investigation of new sources of decoherence; thermal mechanical noise and optical transmission loss have to be analysed. Such an analysis requires adiabatic elimination of the complex transducer dynamics since the Hilbert space dimension is too large to allow numerical simulations. Compared to earlier proposals of optomechanical transducers, our strategy requires no time-dependent control. This simplicity leads to modest requirements on the system parameters; optomechanical cooperativity moderately larger than unity is sufficient and large transmission losses can be tolerated. The approach is scalable to generation of multipartite entanglement and represents a crucial step towards quantum networks with superconducting circuits.

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- 1. Measurement-induced long-distance entanglement of superconducting qubits using optomechanical transducers Ondřej Černotík and Klemens Hammerer Leibniz Universität Hannover Erice, 1 August 2016 PRA 94, 012340
- 2. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ Superconducting systems are among the best candidates for quantum computers. 2 • Controlling microwave ﬁelds with qubits Hofheinz et al., Nature 454, 310 (2008); Nature 459, 546 (2009) • Feedback control of qubits Ristè et al., PRL 109, 240502 (2012); Vijay et al., Nature 490, 77 (2012); de Lange et al., PRL 112, 080501 (2014) • Entanglement generation Ristè et al., Nature 502, 350 (2013); Roch et al., PRL 112, 170501 (2014); Saira et al., PRL 112, 070502 (2014) • Quantum error correction Córcoles et al., Nature Commun. 6, 6979 (2015), Kelly et al., Nature 519, 66 (2015), Ristè et al., Nature Commun. 6, 6983 (2015) R. Schoelkopf
- 3. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ Entanglement between two qubits can be generated by measurement and postselection. 3 C. Hutchison et al., Canadian J. Phys. 87, 225 (2009) N. Roch et al., PRL 112, 170501 (2014) Hint = za† aDispersive coupling |11i |00i |01i + |10i
- 4. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ We want to extend the distance over which the qubits become entangled. 4 - Other proposals: K. Stannigel et al., PRL 105, 220501 (2010) B. Clader, PRA 90, 012324 (2014) Z. Yin et al., PRA 91, 012333 (2015) Experiments: J. Bochmann et al., Nat. Physics 9, 712 (2013) R. Andrews et al., Nat. Physics 10, 321 (2014) T. Bagci et al., Nature 507, 81 (2014) K. Balram et al., Nat. Photon. 10, 346 (2016) ---- 1. Force sensing 2. Equation of motion 3. Feasibility
- 5. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ Optomechanical transducer acts as a force sensor. 5 F = ~ /( p 2xzpf ) S2 F (!) = x2 zpf /[8g2 2 m(!)]Sensitivity: ! ⌧ !m ⌧meas = S2 F (!) F2 = !2 m 16 2g2 ⌧ T1,2Measurement time: H = z(b + b† ) + !mb† b + g(a + a† )(b + b† )
- 6. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ The thermal mechanical bath affects the qubit. 6 mech = S2 f (!) = 2 2 !2 m ¯nDephasing rate: ⌧meas < 1 mech ! C = 4g2 ¯n > 1 2
- 7. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ The system can be modelled using a conditional master equation. 7 D[O]⇢ = O⇢O† 1 2 (O† O⇢ + ⇢O† O) H[O]⇢ = (O hOi)⇢ + ⇢(O† hO† i) H. Wiseman & G. Milburn, Quantum measurement and control (Cambridge) d⇢ = i[H, ⇢]dt + Lq⇢dt + 2X j=1 {(¯n + 1)D[bj] + ¯nD[b† j]}⇢dt + D[a1 a2]⇢dt + p H[i(a1 a2)]⇢dW H = 2X j=1 j z(bj + b† j) + !mb† jbj + g(aj + a† j)(bj + b† j) + i 2 (a1a† 2 a2a† 1)
- 8. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ The transducer is Gaussian and can be adiabatically eliminated. 8 OC et al., PRA 92, 012124 (2015)ˇ 2 qubits Mechanics, light
- 9. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ We obtain an effective equation for the qubits. 9 d⇢q = 2X j=1 1 T1 D[ j ] + ✓ 1 T2 + mech ◆ D[ j z] ⇢qdt + measD[ 1 z + 2 z]⇢qdt + p measH[ 1 z + 2 z]⇢qdW meas = 16 2 g2 !2 m , mech = 2 !2 m (2¯n + 1)
- 10. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ Optical losses introduce additional dephasing. 10 p ⌘ measH[ 1 z + 2 z]⇢q (1 ⌧) measD[ 1 z]⇢q
- 11. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ A transmon qubit can capacitively couple to a nanobeam oscillator. 11 G. Anetsberger et al., Nature Phys. 5, 909 (2009) J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015) = 2⇡ ⇥ 5.8 MHz g = 2⇡ ⇥ 900 kHz = 2⇡ ⇥ 39MHz !m = 2⇡ ⇥ 8.7 MHz Qm = 5 ⇥ 104 T = 20 mK ¯n = 48 T1,2 = 20 µs C = 10 ⌘ Psucc Psucc
- 12. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ The mechanical oscillator can be a membrane. 12 R. Andrews et al., Nature Phys. 10, 312 (2014) T. Bagci et al., Nature 507, 81 (2014) J. Pirkkalainen et al., Nature 494, 211 (2013)
- 13. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ Mechanical oscillators can mediate interaction between light and SC qubits. 13 OC & K. Hammerer, PRA 94, 012340ˇ - C = 4g2 ¯n > 1 2 • Strong optomechanical cooperativity, • Sufﬁcient qubit lifetime