On the first experimental realization of a quantum state feedback

Transcription

1 On the first experimental realization of a quantum state feedback 55th IEEE Conference on Decision and Control Las Vegas, USA, December 12-14, 2016 Pierre Rouchon Centre Automatique et Systèmes, Mines ParisTech, PSL Research University Quantic Research Team, Inria 1 / 32

2 Technologies for quantum simulation and computation 1 Superconduc ng circuits Quantum dots Ultra-cold neutral/rydberg atoms IBM Photons Pe a Trapped ions S. Kuhr OBrien Bla & Wineland Requirement: Scalable modular architecture Control software from the very beginning 1 Courtesy of Walter Riess, IBM Research - Zurich. 2 / 32

3 Nobel Prize in Physics 2012 Serge Haroche David J. Wineland " This year s Nobel Prize in Physics honours the experimental inventions and discoveries that have allowed the measurement and control of individual quantum systems. They belong to two separate but related technologies: ions in a harmonic trap and photons in a cavity" From the Scientific Background on the Nobel Prize in Physics 2012 compiled by the Class for Physics of the Royal Swedish Academy of Sciences, October 9, / 32

4 Outline The photon-box experiment of the Haroche group Modelling based on three quantum rules Measuring photons without destroying them Stabilizing measurement-based feedback Model structures (decoherence, measurement back-action) Discrete-time models (hidden Markov chain) Diffusive models (Wiener process) Jump models (Poisson process) Feedback for quantum systems Measurement-based feedback Coherent (autonomous) feedback and dissipation engineering 4 / 32

5 The first experimental realization of a quantum-state feedback microwave photons (10 GHz) Experiment: C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.M. Raimond, S. Haroche: Real-time quantum feedback prepares and stabilizes photon number states. Nature, 2011, 477, Theory: I. Dotsenko, M. Mirrahimi, M. Brune, S. Haroche, J.M. Raimond, P. Rouchon: Quantum feedback by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states. Physical Review A, 2009, 80: M. Mirrahimi et al. CDC 2009, , H. Amini et al. IEEE Trans. Automatic Control, 57 (8): , R. Somaraju et al., Rev. Math. Phys., 25, , H. Amini et. al., Automatica, 49 (9): , / 32

6 Three quantum features emphasized by the photon box 2 1. Schrödinger: wave funct. i of norm one in Hilbert space H d dt i = i ~ H i, H = H 0 + uh 1, u classical control input Dirac notations: i, i = h, H = H. 2. Origin of dissipation: collapse of the wave packet induced by the measurement of observable O with spectral decomp. P y yp y : I measured output y with probability Py = h P y i; I back-action: P y i i 7! i + = p h Py i 3. Tensor product for composite system (S, M): I Hilbert space H = HS H M I Hamiltonian H = H S I M + H int + I S H M I observable on sub-system M only: O = I S O M. 2 S. Haroche and J.M. Raimond. Exploring the Quantum: Atoms, Cavities and Photons. Oxford Graduate Texts, / 32

7 Composite system (S, M): harmonic oscillator qubit. I System S corresponds to a quantized harmonic oscillator: ( 1 ) X H S = x n ni (x n ) 1 n=0 2 l2 (C), n=0 where ni is the photon-number state with n photons (hn 1 n 2 i = n1,n 2 ). I Meter M is a qubit, a 2-level system: H M = x g gi + x e ei x g, x e 2 C, where gi (resp. ei) is the ground (resp. excited) state (hg gi = he ei = 1 and hg ei = 0) I State of the composite system i 2H S H M : i = +1X n=0 x ng ni gi + x ne ni ei, x ne, x ng 2 C. Ortho-normal basis: ni gi, ni ei n2n. 7 / 32

8 The hidden Markov chain (1) R 1 R 2 C D B B R 2 I When atom comes out B, the quantum state i B of the composite system is separable: i B = i gi. I Just before the measurement in D, the state is in general entangled (not separable): i R2 = U SM i gi = M g i gi + M e i ei where U SM is a unitary transformation (Schrödinger propagator) defining measurement operators M g and M e on H S. U SM unitary implies M gm g + M em e I. 8 / 32

9 The hidden Markov chain (2) Just before the atom detector D the quantum state is entangled: i R2 = M g i gi + M e i ei Just after outcome 3 y 2{g, e}, the state becomes separable:! i D = M y q h M y M y i i Hidden Markov chain: i gi! yi with probability h M ym y i. M y q h M y M y i i! yi k+1 i = 8 < : M g q h k M g M g k i M e k i, y k = g with probability h k M gm g k i; ph k M e M e k i ki, y k = e with probability h k M em e k i; with state k i and output y k 2{g, e} at time-step k: 3 Measurement operator O = I S ( eihe gihg ). 9 / 32

10 Why density operators instead of wave functions i Assume known 0 i and detector out of order: what about 1 i? I Expectation value of 1 ih 1 knowing 0 i: 4 E 1 ih 1 {z } 1 0 i = M g 0 ih 0 {z } 0 Mg + M e 0 ih 0 {z } 0 I Set K ( ), M g M g + M e M e for any operator. I k expectation of k ih k knowing 0 = 0 ih 0 : 1 = K ( 0 ), 2 = K ( 1 ),..., k = K ( k 1 ). Me. Linear map K : positivity and trace preserving (M gm g + M em e = I) called Kraus map or quantum channel. Density operators : Hermitian non-negative operators of trace one. 4 ih : orthogonal projector on line spanned by unitary vector i. 10 / 32

11 The Markov chain with as hidden state Detector efficiency 2 [0, 1]. Measured output y 2{g, e,?}: 8 K g ( k ) Tr K g ( k ), y k = g with probability Tr K g ( k ) ; >< K k+1 = e ( k ) Tr (K e ( k )), y k = e with probability Tr (K e ( k )); K >:? ( k ) Tr (K? ( k )), y k =? with probability Tr (K? ( k )); with Kraus maps We still have: K g ( ) = M g M g, K e ( ) = M e M e K? ( ) =(1 ) M g M g + M e M e. E k+1 k, K ( k )=M g k M g + M e k M e = X y K y ( k ). Imperfections, decoherence modeled similarly 11 / 32

12 A controlled Markov process (input u, hidden state, output y) Input u: classical amplitude of a coherent micro-wave pulse State : the density operator of the photon(s) trapped in the cavity Output y: measurement of the probe atom k+1 = 8 >< D uk K g ( k )D u k Tr K g ( k ),y k = g with probability Tr K g ( k ) ; D uk K e ( k )D u k Tr (K e ( k )),y k = e with probability Tr (K e ( k )); >: D uk K? ( k )D u k Tr (K? ( k )),y k =? with probability Tr (K? ( k )); Controlled displacement unitary operator (u 2 R): D u = e u (a a) with a = upper diag( p 1, p 2,...) the photon annihilation operator. Measurement Kraus operators with linear dispersive interaction M g = cos and M e = sin : M gm g + M em e = I with 0N+ R 2 0N+ R 2 N = a a = diag(0, 1, 2,...) the photon number operator. 12 / 32

13 Open-loop dynamics (u = 0): experimental data y diag(ρ) ρ 13 / 32

14 u = 0: Quantum Non Demolition (QND) measurement of photons. k+1 = 8 >< >: K g ( k ) Tr(K g ( k )),y k = g with probability Tr K g ( k ) ; K e ( k ) Tr(K e ( k )),y k = e with probability Tr (K e ( k )); K? ( k ) Tr(K? ( k )),y k =? with probability Tr (K? ( k )); Photon-number state nih n is a steady-state: K y ( nih n ) / nih n. Martingales W g ( ) = Tr (g(n) ) for any function g: Convergence: W ( ) =1 E W g ( k+1 ) / k = W g ( k ). P n hn ni2 super-martingale, E W ( k+1 ) / k = W ( k ) Q( k ) where Q( ) 0 and Q( ) =0 iff 9 n such that = nih n. Probability to converge to nih n : Tr( nih n 0 )=h n 0 ni. 14 / 32

15 Feedback stabilization around 3-photon state: experimental data y u diag(ρ) ρ 15 / 32

16 Structure of the stabilizing quantum-state feedback scheme With a sampling time of 80 µs, the controller is classical I Goal: stabilization towards nih n. I Step k 1 to step k 1. read y k 1 the measurement outcome for probe atom k compute k from k 1 via k = D u k 1 K yk 1 ( k 1 )D u k 1 Tr K yk 1 ( k 1 ) 3. compute u k as a function of k (state feedback). 4. apply the micro-wave pulse of amplitude u k. Observer/controller structure: 1. real-time state estimation: the quantum Belavkin filter 2. u = f ( ) based on a strict control Lyapunov function V ( ) derived from open-loop martingales Tr (g(n) ): E V ( k+1 ) k, u k = f ( k ) = V ( k ) Q( k ), where Q( ) 0 and Q( ) =0 iff = nih n. 16 / 32

17 Experimental closed-loop data C. Sayrin et. al., Nature 477, 73-77, Sept Decoherence due to finite photon life-time (70 ms) Detection efficiency 40% Detection error rate 10% Delay 4 sampling periods y V u Stabilization around 3-photon state The quantum filter includes decoherence, detector imperfections and delays (Bayes law). Truncation to 9 photons 17 / 32

18 Outline The photon-box experiment of the Haroche group Modelling based on three quantum rules Measuring photons without destroying them Stabilizing measurement-based feedback Model structures (decoherence, measurement back-action) Discrete-time models (hidden Markov chain) Diffusive models (Wiener process) Jump models (Poisson process) Feedback for quantum systems Measurement-based feedback Coherent (autonomous) feedback and dissipation engineering 18 / 32

19 Discrete-time Stochastic Master Equations (SME) Trace preserving Kraus map K u depending on u: X M u, M u, with M u, M u, = I. K u ( ) = X Take a left stochastic matrix y, and set K u,y ( ) = P y, M u, M u,. Associated Markov chain reads: k+1 = K u k,y k ( k ) Tr (K uk,y k ( k )) with y k with probability Tr (K uk,y k ( k )). Classical input u, hidden state, measured output y. Ensemble average given by K u since E k+1 k, u k = K uk ( k ). Markov model useful for: 1. Monte-Carlo simulations of quantum trajectories (decoherence, measurement back-action). 2. quantum filtering to get the quantum state k from 0 and (y 0,...,y k 1 ) (Belavkin quantum filter developed for diffusive models). 3. feedback design and Monte-Carlo closed-loop simulations. 19 / 32

20 Stochastic master equation driven by Wiener process Lindblad equation (H Hermitian, L arbitrary, depending on u): For any t d dt, L( ) = i ~ [H, ] {z } Schrödinger + L L 1 2 (L L + L L) {z }, D L ( ) decoherence 0, 0 7! e tl ( 0 )= t is a trace preserving Kraus map. Continuous-time output t 7! y t with dy t = p Tr (L + L ) t dt + dw t : i d t = [H, ~ t]+l t L 1 2 (L L t + t L L) dt + p L t + t L Tr (L + L ) t t dw t driven by Wiener process dw t ( detection efficiency). Another formulation with Itō differentiation rule: t+dt = K dy t ( ) Tr (K dyt ( )) with proba. density Tr (K dy t ( )) for dy t K dy ( ) =e dy2 2dt M dy M dy +(1 )L tl dt M dy = I + i ~ H 1 2 L L dt + p dyl. 20 / 32

21 Stochastic master equation driven by Poisson process Lindblad equation: d dt, L( ) = i ~ [H, ]+L L 1 2 (L L + L L) Counter t 7! y(t) 2 N (detection imperfections # 0, 2 [0, 1]): dy(t) =0 or 1 : E dy(t) t = # + Tr L t L dt Dynamics: d t = i [H, ~ t]+l t L 1 2 (L L t + t L L) + # t + L t L # + Tr L t L t! dy(t) dt # + Tr L t L dt dy(t) =0: t+dt = K 0( t ) Tr (K 0 ( t )) with probability Tr (K 0( t )) dy(t) =1: t+dt = K 1( t ) Tr (K 1 ( t )) with probability Tr (K 1( t )) K 0 ( ) = M 0 M 0 +(1 )L L dt (1 #dt), K 1 ( ) = # + L L dt M 0 = I + i ~ H 1 2 L L dt 21 / 32

22 Outline The photon-box experiment of the Haroche group Modelling based on three quantum rules Measuring photons without destroying them Stabilizing measurement-based feedback Model structures (decoherence, measurement back-action) Discrete-time models (hidden Markov chain) Diffusive models (Wiener process) Jump models (Poisson process) Feedback for quantum systems Measurement-based feedback Coherent (autonomous) feedback and dissipation engineering 22 / 32

23 Measurement-based feedback u system decoherence P-controller (Markovian feedback a ): for u(t) =f (y(t)), average closed-loop dynamics of remains governed by a Lindblad master equation. PID controller: no Lindblad master equation in closed-loop; classical world quantum world controller y Nonlinear hidden-state stochastic systems: convergence analysis, Lyapunov exponents, dynamic output feedback, delays, robustness,... a H.M. Wiseman: Quantum Trajectories and Feedback. PhD Thesis, University of Queensland, Short sampling times limit feedback complexity 23 / 32

24 First SISO measurement-based feedback for a superconducting qubit 5 d Feedback circuit X*Y 10 MHz Phase error = Feedback signal Rabi reference 3.0 MHz Analog Multiplier Y X a Signal generation Rabi drive c Homodyne setup I LO Readout drive LO Q Digitizer/ computer Q RF RF I b Dilution refrigerator Transmon qubit IN (x2) Paramp HEMT OUT Q 0> 0> Q Readout cavity 1> I 1> I 5 R. Vijay,..., I. Siddiqi. Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77-80, October / 32

25 First MIMO measurement-based feedback for a superconducting qubit 6 a), 2 outputs in 3 inputs out JPC Rabi FM b) Rabi Drift c) FM ~ ~ ~ ~ ~ 6 P. Campagne-Ibarcq,..., B. Huard: Using Spontaneous Emission of a Qubit as a Resource for Feedback Control. Phys. Rev. Lett. 117(6), / 32

26 Measurement-based feedback stabilization of 1 p2 ( gi ei + ei gi) 7 Alice Bob 7 D. Ristè,..., L. DiCarlo: Deterministic entanglement of superconducting qubits by parity measurement and feedback. Nature 502, (2013). 26 / 32

27 Coherent (autonomous) feedback (dissipation engineering) Quantum analogue of Watt speed governor: a dissipative mechanical system controls another mechanical system 8 classical world Optical pumping (Kastler 1950), coherent population trapping (Arimondo 1996) u c system decoherence y c Dissipation engineering, autonomous feedback: (Zoller, Cirac, Wolf, Verstraete, Devoret, Schoelkopf, Siddiqi, Lloyd, Viola, Ticozzi, Mirrahimi, Sarlette,...) u decoherence quantum world controller y (S,L,H) theory and linear quantum systems: quantum feedback networks based on stochastic Schrödinger equation, Heisenberg picture (Gardiner, Yurke, Mabuchi, Genoni, Serafini, Milburn, Wiseman, Doherty, Gough, James, Petersen, Nurdin, Yamamoto, Zhang, Dong,... ) Stability analysis: Kraus maps and Lindblad propagators are always contractions (non commutative diffusion and consensus). 8 J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, / 32

28 Reservoir engineering to stabilize "Schrödinger cats" i + - i 9 atom (controller) R1 Cavity mode (system) Aim: engineer atom-mode interaction, to stabilize -α + α R2 Box of atoms ENS experiment Jaynes-Cumming Hamiltionian DC field: (controls atom frequency) H(t)/~ =! c a a I M +! q (t)i S z /2 + i (t) a - a + /2 with the open-loop control t 7!! q (t) combining dispersive! q 6=! c and resonant! q =! c interactions. Key issues: convergence of k+1 = K ( k )=M g k M g + M e k M e. 9 A. Sarlette et al: Stabilization of nonclassical states of the radiation field in a cavity by reservoir engineering. Phys. Rev. Lett. 107(1), / 32

29 Autonomous feedback stabilization of 1 p2 ( gi ei ei gi) 10 Lindblad master equation: with d dt = + 1 T A T B 1 H(t)/~ = + 2 c cos i[h(t), ]+appled a( ) D A ( )+ 1 2T A D A z ( ) D B ( )+ 1 2T B D B z ( ) A 2 A z + A+ B 2 t + Ax 0 + x B + n e in A + B 2 t ( + A 2 B B z a a a + a B +)+h.c. 10 S. Shankar,..., M.H. Devoret. Autonomously stabilized entanglement between two superconducting quantum bits. Nature, 504: , / 32

30 Combining measurement-based feedback with dissipation engineering Y. Liu,..., M.H. Devoret: Comparing and combining measurement-based and driven-dissipative entanglement stabilization. Phys. Rev. X 6, / 32

31 Concluding remarks I Three key quantum rules (highlighted by Haroche photon-box) 1. Schrödinger equations defining unitary transformations, 2. partial collapse of the wave packet: dissipation induced by measurement of observables with degenerate spectra, 3. tensor product for composite systems, explain structure of stochastic master equations, illustrate importance of spin/spring models. I Feedback control of coherence and entanglement in composite systems: important issues for quantum computing, simulation, metrology and communication. I Composite systems : curse of dimensionality for quantum networks with delays in coherent feedback loops. Importance of collaborations with experimental quantum physicists for defining relevant control engineering questions 31 / 32

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