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- 1. Novel approaches to optomechanical transduction Ondřej Černotík Max Planck Institute for the Science of Light, Erlangen, Germany Leibniz University Hannover, Germany Aarhus University, 18 April 2018
- 2. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ 2 • Quantum gates and processors L. DiCarlo et al., Nature 460, 240 (2009); ibid. 467, 574 (2010); A. Fedorov et al., Nature 481, 170 (2011) • Quantum simulations A. Houck et al., Nature Physics 8, 292 (2012); R. Barends et al., Nature Commun. 6, 7654 (2015); Y. Salathé et al., PRX 5, 021027 (2015) • Quantum error correction A. Córcoles et al., Nature Commun. 6, 6979 (2015); J. Kelly et al., Nature 519, 66 (2015); D. Ristè et al., Nature Commun. 6, 6983 (2015) • Quantum networks Quantum communication, distributed quantum computing, … Schoelkopf group, Yale Superconducting circuits are among the best candidates for quantum computers.
- 3. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Various systems can serve as an interface between microwaves and light. 3 J. Bochmann et al., Nature Phys. 9, 712 (2013) R. Andrews et al., Nature Phys. 10, 321 (2014) T. Bagci et al., Nature 507, 81 (2014) K.C. Balram et al., Nature Photon. 10, 346 (2016) A. Okada et al., arXiv:1705.04593 K. Takeda et al., arXiv:1706.00532 Mechanical oscillators Electrooptics A. Rueda et al., Optica 3, 597 (2016) Magnons R. Hisatomi et al., PRB 93, 174427 (2016) C. O’Brien et al., PRL 113, 063603 (2014) Spins/spin ensembles
- 4. Novel approaches to optomechanical transduction Entanglement generation - Optomechanical transduction Spatially adiabatic conversion
- 5. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Optomechanical interaction is due to radiation pressure. 5 Cavity frequency: !c(ˆx) ⇡ !c(0) + d!c dx ˆx Coupling strength: g0 = d!c dx xzpf = !c L xzpf Hamiltonian: ˆH = ~!c(ˆx)ˆa† ˆa + ~!m ˆb†ˆb ˆH = ~!cˆa† ˆa + ~!m ˆb†ˆb + ~g0ˆa† ˆa(ˆb + ˆb† ) ˆx = xzpf (ˆb + ˆb† ), xzpf = r ~ 2m!m ˆa ˆx !m !c M. Aspelmeyer, T. Kippenberg, and F. Marquardt, RMP 86, 1391 (2014)
- 6. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Strong coupling can be achieved by suitable driving. 6 Optomechanical coupling is weak Solution: strong optical drive ˆa ! ↵ + ˆa, g0 ! g = g0↵ Interaction Hamiltonian ˆHint ⇡ ~g(ˆa + ˆa† )(ˆb + ˆb† ) g0 ⌧ M. Aspelmeyer, T. Kippenberg, and F. Marquardt, RMP 86, 1391 (2014)
- 7. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Driving frequency is an important control parameter. 7 M. Aspelmeyer, T. Kippenberg, and F. Marquardt, RMP 86, 1391 (2014) !m Red-detuned drive: State swapˆHint = ~g(ˆa†ˆb + ˆb† ˆa) !L = !c !m ! !c!L < !m !m Blue-detuned drive: Squeezing !L = !c + !m ˆHint = ~g(ˆaˆb + ˆa†ˆb† ) !!c !L < !m Resonant drive: Position readout !L = !c ˆHint = ~g(ˆa + ˆa† )(ˆb + ˆb† ) ! !c = !L
- 8. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Optomechanical interaction can be observed in various systems. 8
- 9. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Mechanical oscillators are good transducers for microwaves and light. 9 R. Andrews et al., Nature Phys. 10, 321 (2014) g2 g1 ˆa1 ˆa2 ˆb ˆa2,in ˆa2,out ˆa1,outˆa1,in ˆH = g1(ˆa† 1 ˆb + ˆb† ˆa1) + g2(ˆa† 2 ˆb + ˆb† ˆa2) ˆa1 ˆa2 ˆb ˆa1,in/out ˆa2,in/out
- 10. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Mechanical oscillators are good transducers for microwaves and light. 10 R. Andrews et al., Nature Phys. 10, 321 (2014) g2 g1 ˆa1 ˆa2 ˆb ˆa2,in ˆa2,out ˆa1,outˆa1,in ˆH = g1(ˆa† 1 ˆb + ˆb† ˆa1) + g2(ˆa† 2 ˆb + ˆb† ˆa2) dtˆa(t) = Aˆa(t) + Bˆain(t) ˆaout(t) = Cˆa(t) + Dˆain(t) ˆaout(!) = S(!)ˆain(!) = [D C(A + i!1)B]ˆain(!)
- 11. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Mechanical oscillators are good transducers for microwaves and light. 11 R. Andrews et al., Nature Phys. 10, 321 (2014) g2 g1 ˆa1 ˆa2 ˆb ˆa2,in ˆa2,out ˆa1,outˆa1,in ˆH = g1(ˆa† 1 ˆb + ˆb† ˆa1) + g2(ˆa† 2 ˆb + ˆb† ˆa2) g2 1 1 = g2 2 2 Impedance matching i ⌧ !mResolved-sideband regime Strong cooperativity Ci = 4g2 i i ¯n 1 / g2 1 1 = g2 2 2 ⌧ i
- 12. OC and K. Hammerer, PRA 94, 012340 (2016)ˇ Measurement-induced entanglement of superconducting qubits
- 13. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ We can measure qubits using dispersive interaction with ﬁelds. 13 Dispersive coupling ˆHint = ˆzˆa† ˆa |0i |1i D. Ristè et al., PRL 109, 050507 (2012) amplitude phase
- 14. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ We can measure qubits using dispersive interaction with ﬁelds. 14 C. Hutchison et al., Canadian J. Phys. 87, 225 (2009) N. Roch et al., PRL 112, 170501 (2014) Dispersive coupling ˆHint = ˆzˆa† ˆa |11i |00i |01i + |10i | 0i = (|0i + |1i)(|0i + |1i) amplitude phase
- 15. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ D[ ˆO]ˆ⇢ = ˆOˆ⇢ ˆO† 1 2 ( ˆO† ˆOˆ⇢ + ˆ⇢ ˆO† ˆO) H[ ˆO]ˆ⇢ = ( ˆO h ˆOi)ˆ⇢ + ˆ⇢( ˆO† h ˆO† i) ˆH = 2X j=1 ˆj z(ˆbj + ˆb† j) + !m ˆb† j ˆbj + g(ˆaj + ˆa† j)(ˆbj + ˆb† j) + i 2 (ˆa1ˆa† 2 ˆa2ˆa† 1) dˆ⇢ = i[ ˆH, ˆ⇢]dt + Lq ˆ⇢dt + 2X j=1 {(¯n + 1)D[ˆbj] + ¯nD[ˆb† j]}ˆ⇢dt + D[ˆa1 ˆa2]ˆ⇢dt + p H[i(ˆa1 ˆa2)]ˆ⇢dW Extension to room temperature is possible with optomechanical transducers. 15
- 16. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ ˆa ˆb g Optomechanical transducer can act as a force sensor. 16 F = ~ /( p 2xzpf ) Sensitivity: S2 F (!) = x2 zpf /8g2 2 m(!) Measurement time: ⌧meas = S2 F (!) F2 = !2 m 16 2g2 ⌧ T1,2 ˆH = ˆz(ˆb + ˆb† ) + !m ˆb†ˆb + g(ˆa + ˆa† )(ˆb + ˆb† )
- 17. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ The thermal mechanical bath affects the qubit. 17 Dephasing rate: mech = S2 f (!) = 2 2 !2 m ¯n ⌧meas < 1 mech ! C = 4g2 ¯n > 1 2 ˆa ˆb g ¯n
- 18. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Gaussian systems can be adiabatically eliminated from conditional dynamics. 18 OC, D.V. Vasilyev, and K. Hammerer, PRA 92, 012124 (2015) ˇ ˆHint = ˆsT ˆr operatorsˆrˆs state descriptionx,ˆ⇢ System Transducer ...
- 19. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ meas = 16 2 g2 !2 m , mech = 2 !2 m (2¯n + 1) dˆ⇢q = 2X j=1 1 T1 D[ˆj ] + ✓ 1 T2 + mech ◆ D[ˆj z] ˆ⇢qdt + measD[ˆ1 z + ˆ2 z]ˆ⇢qdt + p measH[ˆ1 z + ˆ2 z]ˆ⇢qdW We can obtain an effective equation for the qubits. 19
- 20. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Optical losses introduce additional dephasing. 20 (1 ⌧) measD[ˆ1 z]ˆ⇢q p ⌘ measH[ˆ1 z + ˆ2 z]ˆ⇢q
- 21. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Entanglement generation is possible with existing technology. 21 G. Anetsberger et al., Nature Phys. 5, 909 (2009) J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015) OC and K. Hammerer, PRA 94, 012340 (2016)ˇ = 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 !m Qm ⌘ 0.2 1.0
- 22. OC, S. Mahmoodian, and K. Hammerer, arXiv:1707.03339ˇ Spatially adiabatic frequency conversion in optomechanical arrays
- 23. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ 23 g1 g2 L. Tian, PRL 108, 153604 (2012) Y.-D. Wang and A.A. Clerk, PRL 108, 153603 (2012) Efﬁcient transduction is possible using adiabatic state transfer. ˆ ˆ ˆ ˆH(t) = g1(t)(ˆc† 1 ˆb + ˆb† ˆc1) + g2(t)(ˆc† 2 ˆb + ˆb† ˆc2) = q g2 1(t) + g2 2(t)[ ˆd† 1(t)ˆb + ˆb† ˆd1(t)] ˆd1(t) = 1 p g2 1 + g2 2 [g1(t)ˆc1 + g2(t)ˆc2] ˆd2(t) = 1 p g2 1 + g2 2 [g2(t)ˆc1 g1(t)ˆc2]
- 24. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ 24 Efﬁcient transduction is possible using adiabatic state transfer. G1 G2 ˆ ˆ ˆ ˆH = vi Z dzˆa† i (z)@zˆai(z) + Z dzGi(z)ˆa† i (z)ˆb(z) + H.c. ˆd1(z) = 1 p G2 1 + G2 2 [G1(z)ˆa1 + G2(z)ˆa2] ˆd2(z) = 1 p G2 1 + G2 2 [G2(z)ˆa1 G1(z)ˆa2]
- 25. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ ˆa(z+ j , !) = Sj(!)ˆa(zj , !) ˆa(L, !) = NY j=1 Sj(!)ˆa(0, !) 25 Adiabatic conversion can be approximated in a transducer array. ˆc1 ˆc2 ˆb ˆ ˆ dˆci dt = i 2 ˆci igi ˆb + p iˆai(zj ) dˆb dt = 2 ˆb ig1ˆc1 ig2ˆc2 + p ˆbin ˆai(z+ j ) = p iˆci ˆai(zj )
- 26. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Conversion bandwidth can be increased by increasing the array size. 26 Conversion probability off resonance p1 / g1g2/!3 ⌧ 1 pN = Np1 / g1g2N/!3 1 transducer transducersN ! = 4 p 2 3 g1g2N !1/3
- 27. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Conversion bandwidth can be increased by increasing the array size. 27 p1 / g1g2/!3 ⌧ 1 pN = Np1 / g1g2N/!3 1 transducer transducersN ! = 4 p 2 3 g1g2N !1/3 10 50 200 N = 1
- 28. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Added noise is reduced by increasing the array size. 28 Single transducer ˆa(z+ j , !) = Sj(!)ˆa(zj , !) + Vj(!) ˆfj Output spectrum S2 out(!) / S2 in(!) + 1 C , C = 4g2 ¯n Many transducers S2 out(!) / S2 in(!) + N C Dark mode ˆd2(z+ j , !) / ˆd2(zj , !) + 1 N Vj(!) ˆfj Total noise S2 add(!) / 1 CN Collective cooperativity Ccoll = 4g2 N ¯n > 1
- 29. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Efﬁcient conversion is possible in presence of optical losses. 29 Propagation loss Backscattering RL ⌘ ⌘ = 0.05 0.01 0.2 0.001 L/R = 0 0.1 0.5 0.9
- 30. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ Optomechanical transducers present a promising route to frequency conversion. 30 Arbitrary input Small bandwidth OC, S. Mahmoodian, and K. Hammerer, arXiv:1707.03339ˇ Large bandwidth, noise resilient Arbitrary signals OC and K. Hammerer, PRA 94, 012340 (2016)ˇ Entanglement generation Builds on existing experiments System Transducer ... OC, D.V. Vasilyev, and K. Hammerer, PRA 92, 012124 (2015)ˇ Gaussian transducers under continuous measurement General recipe for adiabatic elimination