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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 mediated 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. Entanglement generation is possible without ground-state cooling of the mechanical oscillators for systems with optomechanical cooperativity moderately larger than unity; in addition, our setup tolerates a substantial transmission loss. 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 with Optomechanical Transducers Ondřej Černotík and Klemens Hammerer Leibniz Universität Hannover Palacký University Olomouc, 23 September 2015
- 2. Quantum information 2 Processing Superconducting qubits, trapped ions, … Schoelkopf Blatt Transfer Light Zeilinger Storage Solid-state spins, atomic ensembles, mechanical oscillators, … Lehnert Polzik Superconducting qubits
- 3. SC qubits can be interfaced with light using spin ensembles. 3 C. O’Brien et al., PRL 113, 063603 (2014) K. Xia & J. Twamley, PRA 91, 042307 (2015) C. O’Brien et al.
- 4. Mechanical oscillators can also mediate the coupling. 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. Mechanical oscillators can also mediate the coupling. 5 K. Xia et al., Sci. Rep. 4, 5571 (2014) K. Stannigel et al., PRL 105, 220501 (2010)
- 6. Vg Mechanical coupling Single-qubit readout Two-qubit readout Implementations This talk will be about:
- 7. Coupling light and SC qubits to mechanical oscillators Vg
- 8. 8 a x !, ⌦, ¯n !(x) ⇡ !(0) + d! dx x Cavity frequency: g0 = d! dx xzpf = ! L xzpfCoupling strength: xzpf = r ~ 2m⌦ x = xzpf (b + b† ), M . A s p e l m e y e r, T. Kippenberg, F. Marquardt, RMP 86, 1391 (2014) Hamiltonian: H = ~!(x)a† a + ~⌦b† b H = ~!a† a + ~⌦b† b + ~g0a† a(b + b† ) Optomechanical interaction arises due to radiation pressure.
- 9. ⌦ Strong coupling can be achieved using laser driving. 9 Optomechanical coupling is weak g0 = ! xzpf L ⇡ 25 Hz Solution: strong optical drive a ! ↵ + a Interaction Hamiltonian Hint = ~g0↵(a + a† )(b + b† ) M . A s p e l m e y e r, T. Kippenberg, F. Marquardt, RMP 86, 1391 (2014) Red-detuned drive: Hint ⇡ ~g(a† b + b† a) Optomechanical cooling !L = ! ⌦
- 10. Strong coupling can be achieved using laser driving. 10 Optomechanical coupling is weak g0 = ! xzpf L ⇡ 25 Hz Solution: strong optical drive a ! ↵ + a ⌦ Interaction Hamiltonian Hint = ~g0↵(a + a† )(b + b† ) M . A s p e l m e y e r, T. Kippenberg, F. Marquardt, RMP 86, 1391 (2014) Blue-detuned drive: Hint ⇡ ~g(ab + a† b† ) Two-mode squeezing !L = ! + ⌦
- 11. ⌦ Strong coupling can be achieved using laser driving. 11 Optomechanical coupling is weak g0 = ! xzpf L ⇡ 25 Hz Solution: strong optical drive a ! ↵ + a ⌦ Interaction Hamiltonian Hint = ~g0↵(a + a† )(b + b† ) M . A s p e l m e y e r, T. Kippenberg, F. Marquardt, RMP 86, 1391 (2014) Resonant drive: Hint ⇡ ~g(a + a† )(b + b† ) Position readout ! = !L
- 12. ' = '1 '2 Josephson junction is a basic building block of SC circuits. 12 Superconductor Insulator (∼ 1 nm) Superconductor Junction parameters: • critical current , • capacitance , • phase I0 C EJ = ~I0 2e EC = (2e)2 2C Josephson energy charging energy Energy scale: K. Bennemann & J. Ketterson, Superconductivity (Springer) p ⇢ei'1 p ⇢ei'2
- 13. ' = '1 '2 Josephson junction is a basic building block of SC circuits. 13 Superconductor Insulator (∼ 1 nm) Superconductor Junction parameters: • critical current , • capacitance , • phase I0 C V = ~ 2e ˙', I = I0 sin ' Josephson relations ˙I = I0 cos(') ˙' V = ~ 2e 1 I0 cos ' ˙I = L(') ˙I K. Bennemann & J. Ketterson, Superconductivity (Springer) p ⇢ei'1 p ⇢ei'2
- 14. Charge qubit is a voltage-biased JJ. 14 Electrostatic energy: ECoulomb = 4EC(N Ng)2 Ng = CgVg 2e , EC = e2 2C Cg Vg Ng Energy K. Bennemann & J. Ketterson, Superconductivity (Springer)
- 15. Charge qubit is a voltage-biased JJ. 15 Cg Vg Ng Energy Two-level approximation: H = 2EC(2Ng 1) z EJ 2 x Total Hamiltonian: H = 4EC(N Ng)2 + EJ cos ' EC EJ K. Bennemann & J. Ketterson, Superconductivity (Springer)
- 16. Mechanical coupling is achieved using a mechanically compliant capacitor. 16 Charge qubit with a movable gate H = 4EC[N Ng(x)]2 + EJ cos ' + ~⌦b† b Vg x Gate charge: Ng(x) ⇡ CgVg 2e + Vg 2e dCg dx x Hint = 2EC Vg e dCg dx xzpf (b + b† ) z Interaction Hamiltonian: T. Heikkilä et al., PRL 112, 203603 (2014)
- 17. Optical readout of a superconducting qubit
- 18. We can use an optomechanical system to read out the state of a qubit. 18 d⇢ = i[Hint, ⇢]dt + LT ⇢dt + p H[aei ]⇢dW LT ⇢ = i[HT , ⇢] + {(¯n + 1)D[b] + ¯nD[b† ]}⇢ + D[a]⇢ D[O]⇢ = O⇢O† 1 2 (O† O⇢ + ⇢O† O) H[O]⇢ = (O hOi)⇢ + ⇢(O† hO† i) H = z(b + b† ) + !b† b + g(a + a† )(b + b† ) = Hint + HT H. Wiseman & G. Milburn, Quantum measurement and control (Cambridge)
- 19. We adiabatically eliminate the transducer degrees of freedom. 19 d⇢q = ( meas + mech)D[ z]⇢qdt + p measH[ z]⇢qdW meas = 16 2 g2 !2 , mech = 2 !2 (2¯n + 1) Efﬁcient readout: meas mech Optomechanical cooperativity C = 4g2 ¯n 1
- 20. Intermezzo: Adiabatic elimination of Gaussian quantum systems System Transducer System
- 21. Gaussian systems are described by quadratic Hamiltonians, linear jumps, and homodyne measurements. 21 d⇢ = i[H, ⇢]dt + X n D[jn]⇢dt + X m H[ m]⇢dWm H = 1 2 rT Rr, jn = ⇠T n r, m = (cm + imm)T r r = (q1, p1, . . . , qN , pN )T
- 22. Dynamics can be described using statistical moments of canonical operators. 22 Mean values: Covariance matrix: d⇢ = i[H, ⇢]dt + X n D[jn]⇢dt + X m H[ m]⇢dW dx = Axdt + X m ( cm mm)dW ˙ = A + AT + 2N 2 X m ( cm mm)( cm mm)T x = tr{r⇢} ij = tr{[ri, rj]+⇢} 2xixj
- 23. Dynamics can be described using statistical moments of canonical operators. 23 Mean values: Covariance matrix: d⇢ = i[H, ⇢]dt + X n D[jn]⇢dt + X m H[ m]⇢dW dx = Axdt + X m ( cm mm)dW ˙ = A + AT + 2N 2 X m ( cm mm)( cm mm)T x = tr{r⇢} ij = tr{[ri, rj]+⇢} 2xixj
- 24. Description using statistical moments enables adiabatic elimination. 24 OC et al., PRA 92, 012124 (2015)ˇ System Transducer ⇢ x,
- 25. Two-qubit readout using light
- 26. Joint measurement on two qubits can generate entanglement between them. 26 1 z + 2 z | 0i = (|0i + |1i)(|0i + |1i) ! 8 < : |00i |11i |01i + |10i C. Hutchison et al., Canadian J. Phys. 87, 225 (2009) D. Ristè et al., Nature 502, 350 (2013) N. Roch et al., PRL 112, 170501 (2014)
- 27. The system is treated similarly to a single qubit. 27 meas = 16 2 g2 !2 , mech = 2 !2 (2¯n + 1) d⇢q = 1 T1 D[ j ]⇢qdt + ✓ 1 T2 + mech ◆ D[ j z]⇢qdt+ + measD[ 1 z 2 z]⇢qdt + p measH[ 1 z 2 z]⇢qdW
- 28. We can postselect entangled two-qubit states. 28
- 29. We can postselect entangled two-qubit states. 29
- 30. We can also include optical losses. 30
- 31. We can also include optical losses. 31
- 32. Experimental implementations of half-parity measurements
- 33. The mechanical system can be formed by a nanobeam. 33 G. Anetsberger et al., Nature Phys. 5, 909 (2009) J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015)
- 34. The mechanical oscillator can be a membrane. 34 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)
- 35. Mechanical oscillators can also couple to ﬂux qubits. 35 F. Xue et al., NJP 9, 35 (2007)
- 36. Other kinds of qubits can be used as well. 36 P. Rabl et al., PRB 79, 041302 (2009) P. Rabl et al., Nature Phys. 6, 602 (2010) S. Kolkowitz et al., Science 335, 1603 (2012)
- 37. Summary 37 Vg
- 38. Mechanical oscillators can mediate interaction between light and SC qubits. 38 • Strong optomechanical cooperativity, • Sufﬁcient qubit lifetime C = 4g2 ¯n 1
- 39. More complex schemes can be designed. 39 • Measurement feedback • Two-mode optomechanical driving • More experimental implementations