Quantum Reservoir Engineering

Transcription

1 Departments of Physics and Applied Physics, Yale University Quantum Reservoir Engineering Towards Quantum Simulators with Superconducting Qubits SMG Claudia De Grandi (Yale University) Siddiqi Group (Berkeley)

2 Noise and Dissipation: Friend or Foe? Clerk et al. Rev. Mod. Phys. 82, 1155 (2010) 2

3 Challenge: Quantum state control in the presence of noise and dissipation How do we correct non-unitary errors? ρ = = EρE; EE I 3

4 Challenge: Quantum state control in the presence of noise and dissipation How do we correct non-unitary errors? ρ = = EρE; EE I Active feedback: classical measurement result fed back to controller which applies a unitary that is conditioned on measurement result. 4

5 Challenge: Quantum state control in the presence of noise and dissipation How do we correct non-unitary errors? ρ = = EρE; EE I Autonomous feedback: embedded into the system as a quantum bath Paradox: We can use dissipation as a tool to create and maintain coherence. (Poyatos et al. PRL 1996) Interference between coherent drives and quantum noise of bath can be useful. (Murch et al. PRL 2012) 5

6 Q Example: Photon shot noise in a driven damped cavity Coherent drive Interference leads to amplitude fluctuations I 2 1/2 1/2 (nˆ n) = n Vacuum noise Rectification of vacuum noise ˆ = aa (non-linear in field) n Interference κ t 2 2 ntn ˆ( ) ˆ(0) = n + ne Coherent drive Nota Bene Cavity filtered vacuum noise (zero temperature) ω R 0 ω R ω 6

7 Example: Photon shot noise in a driven damped cavity Rectification of vacuum noise when drive is red detuned from cavity Spectral density of shot noise moves up to positive frequencies Bath can absorb energy (by Raman scattering of pump photon up to cavity) Bath cannot emit energy Cooling! V = χ O ( ˆ system n n) shot noise Coherent drive Interference Cavity filtered vacuum noise ω R 0 ω R ω 7

8 Shot noise spectral density for drive red-detuned by distance Clerk et al. Rev. Mod. Phys. 82, 1155 (2010) S nn ( ω) = n κ ( ω ) + ( κ / 2) 2 2 S ( ) nn ω Quantum shot noise spectral density is ASYMMETRIC + κ ω 1 0 Γ01 Γ10 Γ = χ S nn ( ) bath absorbs Γ = 2 01 χ S nn ( ) bath emits Fermi Golden Rule Picture: Clerk et al. Rev. Mod. Phys. 82, 1155 (2010) 8

9 Autonomous Feedback: Quantum Reservoir Engineering We can use photon shot noise to cool a qubit + x = 1 g + e 2 to any point on the Bloch sphere, e.g. ( ) x +x H X Z = Ω ˆ Rabi σ + χσ ( n n) Ω Rabi Rabi drive on qubit <1mK (!) Cavity-assisted quantum bath engineering (Phys. Rev. Lett. 2012) Berkeley: K. W. Murch, S. J. Weber, I. Siddiqi Yale: U. Vool, D. Zhou, SMG shot noise of driven cavity dispersive coupling Dissipator cools qubit via ump operator: c= + x x Two qubit entanglement: S. Shankar et al. (Devoret) Nature (2013). 9

10 Quantum Reservoir Engineering to create persistent Ramsey fringes Dissipator cools qubit via ump operator: c= + x x T eff ~ 150µ K Persistent Ramsey fringes (with enhanced visibility) Murch et al. PRL (2012) 10

11 Canonical Ensemble Quantum Simulator for 1D Bose-Hubbard Model Berkeley: Shay Hacohen-Gourgy Vinay Ramasesh Irfan Siddiqi Yale: Claudia De Grandi SMG Transmon qubit array in cavity 11

12 End qubits flux-tunable (SQUIDs) ω L = 2π x GHz ω R = 2π x GHz α L = α R = - 2π x 214 MHz Cavity ω M = 2π x GHz κ = 2π x 10 MHz g 1 = 2π x GHz g 2 = 2π x GHz g 3 = 2π x GHz Middle qubit fixed (single-unction) ω M = 2π x GHz α M = - 2π x 240 MHz Couplings J 12, J 23 = 2π x GHz J 13 = 2π x GHz 12

13 First 9 excited levels of three-transmon array Dark state F E 2 boson manifold Theory fit J ~ 180MHz α ~ 240MHz 1 boson manifold Transmon qubit array in cavity 13

14 Canonical Ensemble Quantum Simulator for 1D Bose-Hubbard Model Engineer bath to remove entropy and energy but not bosons! [cf. Jake Taylor Chemical Potential for Light ] X X X H bb =, H = ω bb + J bb α bbbb 0 site energy Bose-Hubbard Model k k k 2 boson hopping (dipole coupling) Hubbard U<0 14

15 X X X H = ω aa+ ω bb + J bb + g ( ab + ab) V 0 = Couple Bose-Hubbard lattice to cavity photons to create dissipation 1 2 Q k k k α bbbb N.B. H0, bb 0 15

16 Diagonalize Quadratic H 0 H = ω aa+ ω bb + J bb + g ( ab + ab) k 0 Q k k k a= Λ A+ Λ B 00 0 b = Λ A+ Λ B k 0 k H = ω AA+ ω BB 0 Q 16

17 H ω AA ω BB = + 0 Q Quartic Term becomes (in RWA): V = 1 2 V~ α BBBB+ A A χ BB + χ AAAA ikl α bbbb ikl i k l i i 0 i + Normal modes scatter due to Hubbard interactions H + V BB = 0, 0* Dispersive coupling of (dressed) cavity to normal modes Small cavity self-kerr (Fine print re. Purcell effect) 17

18 Quantum Bath Engineering for 1D Bose-Hubbard Model X X X Red detuned drive Entropy and energy leave via Raman photons at cavity frequency A A χ BB nˆ χ B B i i i i i i = Shot noise fluctuations cause cooling transitions which conserve particle number. 18

19 A A χ BB nˆ χ B B i i i i i i = Shot noise fluctuations cause cooling transitions which conserve particle number. S nn ( ω) = n Fermi Golden Rule cooling rate Γ = Ψ χ ( ) ibb i Ψ Snn ω ω i κ ( ω ) + ( κ / 2) S ( ) nn ω κ Drive detuning + ω 19

20 NATURAL DECAY DYNAMICS 20

21 COOLING COHERENT DRIVES TRANSFER EXCITATIONS BETWEEN SUBSPACES COOLING DRIVES TRANSFER EXCITATIONS WITHIN A SUBSPACE ACCESS DARK STATES 21

22 COOLING RATES 22

23 STATE STABILIZATION: F 1 > Can stabilize any many-body eigenstate Preserves particle number in Bose-Hubbard Model 23

24 PERSPECTIVES STATE SYMMETRY CAN FACILITATE COOLING/QBE COOLING OF QUBIT CHAINS - PREPARE & HOLD N-BODY EIGENSTATE (EG. RESET) - DISSIPATIVE BOSE-HUBBARD PHYSICS - ACCESS DARK STATES - ENERGY FLOW / THERMODYNAMICS - QUANTUM EMULATION IN LONG-CHAINS / ARRAYS FOR FUTURE QUANTUM SIMULATOR NEED: ABILITY TO MEASURE LOCAL QUBIT CORRELATORS IN MANY-BODY STATE 24

25 EXTRA SLIDES BEYOND HERE 27

26 Dissipation and decoherence are caused by coupling to quantum and thermal noise of the surrounding reservoir. ωq z ˆ () x Example: H = σ + Ft σ 2 Classically the noise correlator is real: G () t FtF ˆ() ˆ(0) FF and hence its spectral density iωt S ( ) ˆ( ) ˆ FF ω dte F t F(0) is symmetric in frequency: S FF ( ω) = S ( ω) FF Not true quantum mechanically because: Ft ˆ( ), Fˆ(0) 0 28

27 Example: H ωσ z = q + ˆ () x Ftσ Ft ˆ( ), Fˆ(0) 0 Implies FtF ˆ( ) ˆ(0) is not Hermitian and so the noise correlator G () t FtF ˆ() ˆ(0) FF is complex and hence its spectral density iωt S ( ) ˆ( ) ˆ FF ω dte F t F(0) can be asymmetric in frequency: S FF ( ω) S ( ω) FF 29

28 harmonic oscillator example: ZPF ( ˆ ˆ ) ˆF = F b+ b 2 G ˆ ˆ ˆ ˆ ˆ ˆ FF () t F() t F(0) = F ZPF b() t b (0) + b () t b(0) 2 ( ) iω0t + iω0t GFF () t = F ZPF nb + 1e + ne B (is complex) S FF ( ω) = 2 π ( n ) B + 1 δω ( ω0) + nbδω ( + ω0) Stimulated emission (bath absorbs) Absorption (bath emits) (is asymmetric) 30

29 quantum noise NOT (necessarily) symmetric in frequency: S ( ) FF ω ω < 0 bath emits energy into system ω > 0 bath absorbs energy from system Ω +Ω ω In thermal equilibrium: (or pseudo-equilibrium) S S FF FF ( +Ω ) = ( Ω) e β Ω = 1 for 0 31

30 Fermi s Golden Rule in terms of noise spectral density Example: H ωq = σ z + ˆ () x Ft σ Γ01 Γ10 Γ = S ( + ω ) bath absorbs 10 FF q Γ = S ( ω ) bath emits 01 FF q For quantum noise picture of FGR see: Clerk et al. Rev. Mod. Phys. 82, 1155 (2010) 32

31 Shot noise as rectified vacuum noise Rectified vacuum noise (number fluctuations) (mostly positive freq.) classical laser drive (pos./neg. frequency) Vacuum noise in cavity (electric field, no photons) (positive frequency only) ω L 0 + Electric field spectral density Photon number spectral density S ( ) EE ω S ( ) nn ω κ +ω L ω R κ ω R ω L ω 33

32 Rabi experiment: decoherence of driven qubit in rotating frame: ΩR 1 x () () H= σ + Ut ξ t σ U () t 2 2 G. Ithier, et al., Phys. Rev. B 72, (2005) = + T 4T 2 T ( Ω ) T Rabi 1 ϕ R ϕ 1 1 { } 1 = S zz ( +Ω R) + S zz ( ΩR) ( Ω ) 4 T R but can be smaller if Rabi rate is high ϕ Ω R S zz ( ω) +Ω R ΩR kt B noise is classical so symmetric in frequency (and probably non-eq.) ω 34

33 Choose cavity drive detuning to match mechanical oscillator frequency ω =Ω R ω L M Resolved sideband limit: +Ω M > κ noise is cold Shot noise emits energy into oscillator ω < 0 S ( ) nn ω Shot noise (radiation pressure) absorbs energy from oscillator κ ω > 0 Ω M +Ω M ω 35

34 Fermi s Golden Rule in terms of noise spectral density H X Z = Ω ˆ Rabi σ + χσ ( n n) shot noise of driven cavity X +X Γ01 Γ10 Γ = +Ω 2 10 χ S nn ( R ) bath absorbs Γ = Ω 2 01 χ S nn ( R ) bath emits For quantum noise picture of FGR see: Clerk et al. Rev. Mod. Phys. 82, 1155 (2010) 36