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- 1. Measurement-induced long-distance entanglement of superconducting qubits using optomechanical transducers Ondřej Černotík and Klemens Hammerer Leibniz Universität Hannover, Germany GRS Ventura, 5 March 2016 arXiv:1512.00768
- 2. Cernotík (Hannover): Entanglement of superconducting qubitsˇ 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) • 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) • Entanglement generation Ristè et al., Nature 502, 350 (2013); Roch et al., PRL 112, 170501 (2014); Saira et al., PRL 112, 070502 (2014) Schoelkopf
- 3. Cernotík (Hannover): Entanglement of superconducting qubitsˇ 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. Cernotík (Hannover): Entanglement of superconducting qubitsˇ Mechanical oscillators can mediate coupling between microwaves and light. 4 T. Bagci et al., Nature 507, 81 (2014)R. Andrews et al., Nature Phys. 10, 321 (2014) Z. Yin et al., PRA 91, 012333 (2015)
- 5. Cernotík (Hannover): Entanglement of superconducting qubitsˇ Mechanical oscillators can mediate coupling between microwaves and light. 5 K. Xia et al., Sci. Rep. 4, 5571 (2014) K. Stannigel et al., PRL 105, 220501 (2010)
- 6. Cernotík (Hannover): Entanglement of superconducting qubitsˇ Optomechanical transducer acts as a force sensor. 6 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† )
- 7. Cernotík (Hannover): Entanglement of superconducting qubitsˇ The thermal mechanical bath affects the qubit. 7 mech = S2 f (!) = 2 2 !2 m ¯nDephasing rate: ⌧meas < 1 mech ! C = 4g2 ¯n > 1 2
- 8. Cernotík (Hannover): Entanglement of superconducting qubitsˇ The system can be modelled using a conditional master equation. 8 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)
- 9. Cernotík (Hannover): Entanglement of superconducting qubitsˇ The transducer is Gaussian and can be adiabatically eliminated. 9 OC et al., PRA 92, 012124 (2015)ˇ 2 qubits Mechanics, light
- 10. Cernotík (Hannover): Entanglement of superconducting qubitsˇ We obtain an effective equation for the qubits. 10 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)
- 11. Cernotík (Hannover): Entanglement of superconducting qubitsˇ Optical losses introduce additional dephasing. 11 d⇢q = 2X j=1 ✓ 1 T1 D[ j ] + 1 T2 D[ j z] ◆ ⇢qdt + [(1 ⌧) meas + mech]D[ 1 z]⇢qdt + ⌧ mechD[ 2 z]⇢qdt + ⌧ measD[ 1 z + 2 z]⇢qdt + p ⌧⌘ measH[ 1 z + 2 z]⇢qdW
- 12. Cernotík (Hannover): Entanglement of superconducting qubitsˇ A transmon qubit can capacitively couple to a nanobeam oscillator. 12 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
- 13. Cernotík (Hannover): Entanglement of superconducting qubitsˇ The mechanical oscillator can also be formed by a membrane. 13 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)
- 14. Cernotík (Hannover): Entanglement of superconducting qubitsˇ Mechanical oscillators can mediate interaction between light and SC qubits. 14 • Strong optomechanical cooperativity, • Sufﬁcient qubit lifetime OC & K. Hammerer, arXiv:1512.00768ˇ - C = 4g2 ¯n > 1 2
- 15. Cernotík (Hannover): Entanglement of superconducting qubitsˇ More complex schemes can be designed. 15 • Using a microwave cavity • Two-mode optomechanical driving • Measurement-based feedback • More experimental implementations OC & K. Hammerer, arXiv:1512.00768ˇ -