Coherent Quantum Noise Cancellation for Opto-mechanical Sensors

Transcription

1 Coherent Quantum Noise Cancellation for Opto-mechanical Sensors Mankei Tsang and Carlton M. Caves Center for Quantum Information and Control, UNM Keck Foundation Center for Extreme Quantum Information Theory, MIT Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.1/32

2 Quantum Sensing Kominis et al., Nature 422, 596 (2003). Rugar et al., Nature 430, 329 (2004). O Connell et al., 464, 697 (2010). Quantum Estimation Quantum Control Wildermuth et al., Nature 435, 440 (2005). Kippenberg and Vahala, Science 321, 1172 (2008), and references therein. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.2/32

3 Outline Quantum Bayesian Estimation Filtering Smoothing Optical Phase-Locked Loops Opto-mechanical Sensors Tsang, Shapiro, and Lloyd, PRA 78, (2008); PRA 79, (2009); Tsang, PRL 102, (2009); PRA 80, (2009); Tsang, PRA 81, (2010). Quantum Noise Cancellation (QNC) Flowcharts QNC for Opto-mechanical Sensors Tsang and Caves, under preparation. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.3/32

4 Tracking an Aircraft or submarine, terrorist, criminal, mosquito, cancer cell, nanoparticle,... Use Bayes theorem: P(x t y t ) = P(y t x t )P(x t ) R dxt (numerator), (1) P(y t x t ) from observation noise, P(x t ) from a priori information. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.4/32

5 Prediction Assume x t is a Markov process, use Chapman-Kolmogorov equation: P(x t+δt y t ) = Z dx t P(x t+δt x t )P(x t y t ) (2) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.5/32

6 Filtering: Real-Time Estimation Applying Bayes theorem and Chapman-Kolmogorov equation repeatedly, we can obtain P(x t y t δt,..., y t0 +δt, y t0 ) (3) Useful for control, weather and finance forecast, etc. Wiener, Stratonovich, Kalman, Kushner, etc. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.6/32

7 Quantum Filtering Use quantum Bayes theorem, ˆρ t ( y t ) = ˆM(y t )ˆρ t ˆM (y t ) tr(numerator) (4) Use a completely positive map to evolve the system state (analogous to Chapman-Kolmogorov), ˆρ t+δt ( y t ) = X µ ˆK µˆρ t ( y t ) ˆK µ (5) Belavkin, Barchielli, Carmichael, Caves, Milburn, Wiseman, Mabuchi, etc. Useful for cavity QED, squeezing by QND measurement and feedback, etc. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.7/32

8 Hybrid Classical-Quantum Filtering Define hybrid density operator ˆρ(x t ), P(x t ) = tr[ˆρ(x t )], ˆρ t = R dx tˆρ(x t ) Use generalized quantum Bayes theorem and positive map + Chapman-Kolmogorov: ˆρ(x t y t ) = ˆM(y t x t )ˆρ(x t ) ˆM (y t x t ) R, (6) dxt tr(numerator) ˆρ(x t+δt y t ) = Z dx t P(x t+δt x t ) X µ ˆK µ (x t )ˆρ(x t y t ) ˆK µ(x t ) (7) Tsang, PRL 102, (2009). Parameter Estimation: Chase and Geremia, PRA 79, (2009). Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.8/32

9 Hybrid Filtering Equations Hybrid Belavkin equation (analogous to Kushner-Stratonovich):» dˆρ(x, t) = dt L 0 + L(x) A µ + B µν ˆρ(x, t) x µ x µ x ν + dt h 2ĈT R 1ˆρ(x, i t)ĉ Ĉ T R 1 Ĉˆρ(x, t) ˆρ(x, t)ĉ T R 1 Ĉ dy t dt 2 DĈ + Ĉ E«T R 1 h Ĉ Ĉ i ˆρ(x, t) + H.c. (8) Linear version (analogous to Duncan-Mortensen-Zakai):» d ˆf(x, t) = dt L 0 + L(x) A µ + B µν ˆf(x, t) x µ x µ x ν + dt h 8 2ĈT R 1 ˆf(x, t) Ĉ Ĉ T R 1 Ĉˆρ(x, t) ˆf(x, i t)ĉ T R 1 Ĉ + 1 h 2 dyt t R 1 Ĉ ˆf(x, i ˆf(x, t) t) + H.c., ˆρ(x, t) = R. (9) dx tr[ ˆf(x, t)] Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.9/32

10 Smoothing: Estimation with Delay More accurate than filtering Sensing, analog communication, astronomy, crime investigation,... Delay Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.10/32

11 Quantum Smoothing? Conventional quantum theory is a predictive theory Quantum state described by Ψ t or ˆρ t can only be conditioned only upon past observations Weak values by Aharonov, Vaidman, et al. Quantum retrodiction by Barnett, Pegg, Yanagisawa, etc. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.11/32

12 Hybrid Time-Symmetric Smoothing Use two operators to describe system: density operator ˆρ(x t y past ) and a retrodictive effect operator Ê(y future x t ) P(x t y past, y future ) = i tr hê(yfuture x t )ˆρ(x t y past ) R (10) dx(numerator) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.12/32

13 Equations Solve the predictive equation from t 0 to t using a priori ˆf as initial condition, and solve the retrodictive equation from T to t using ĝ(x, T) ˆ1 as the final condition d ˆf = dtl(x) ˆf + dt 8 dĝ = dtl (x)ĝ + dt 8 2ĈT R 1 ˆfĈ Ĉ T 1Ĉ R ˆf ˆfĈ! T 1Ĉ R + 1 dy T t R 1 Ĉ ˆf + ˆfĈ 2 2Ĉ T R 1 ĝĉ Ĉ T R 1 Ĉĝ ĝĉ T R 1 Ĉ h(x, t) = P(x t = x y past, y future ) =! + 1 dy T R 1 Ĉ ĝ + ĝĉ 2 h tr ĝ(x, t) ˆf(x, i t) R dx(numerator) Tsang, PRL 102, (2009). Classical: Pardoux, Stochastics 6, 193 (1982). Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.13/32

14 Phase-Locked Loop Personick, IEEE Trans. Inform. Th. IT-17, 240 (1971); Wiseman, PRL 75, 4587 (1995); Armen et al., PRL 89, (2002); Berry and Wiseman, PRA 65, (2002); 73, (2006); Tsang, Shapiro, and Lloyd, PRA 78, (2008); 79, (2009); Tsang, PRA 80, (2009). Wheatley et al., PRL 104, (2010). Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.14/32

15 Opto-mechanical Force Sensor Hamiltonian: Ĥ(x t ) = ˆp2 2m x tˆq (11) Caves et al., RMP 52, 341 (1980); Braginsky and Khalili, Quantum Measurements (Cambridge University Press, Cambridge, 1992); Mabuchi, PRA 58, 123 (1998); Verstraete et al., PRA 64, (2001). Assume force is an Ornstein-Uhlenbeck process: dx t = ax t dt + bdw t (12) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.15/32

16 Filtering vs Smoothing at Steady State a/ b = 0.01/( m) 1/4, G = 3/2 γ/m 1/2 b, s qq = p mγ/ Σ qq, s qp = Σ qp /, s pp = Σ pp / p 3 mγ, blue: filtering, green: x t = 0, red: smoothing 6 4 Position Variance s qq s qq (G ) π qq Momentum Variance s pp s pp (G ) π pp db db G G Uncertainty Product 4(s qq s pp -s 2 qp) Heisenberg 4π qq π pp Signal-to-Noise Ratio filtering smoothing db 5 db G G Signal-to-Noise Ratio can be further improved by frequency-dependent squeezing and passing the output light through some detuned cavities [Kimble et al., PRD 65, Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.16/32

17 Opto-mechanical Sensor dq dt = p m, du dt = κ 2 u + r κ 2 ξ, dv dt = γq κ 2 v + dp dt = mω2 mq + f + γu, (13) r κ 2 η, (14) U = 2κu ξ, V = 2κv η. (15) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.17/32

18 Flowchart Representation dq dt = p m, du dt = κ 2 u + r κ 2 ξ, dv dt = γq κ 2 v + dp dt = mω2 mq + f + γu, (16) r κ 2 η, (17) U = 2κu ξ, V = 2κv η. (18) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.18/32

19 Noise Cancellation Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.19/32

20 Quantum Noise Cancellation Coherent Feedforward Quantum Control Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.20/32

21 Modifying Input/Output Optics Kimble et al., PRD 65, (2001) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.21/32

22 Summary Quantum Bayesian Estimation Filtering Smoothing Optical Phase-Locked Loops Opto-mechanical Sensors Tsang, Shapiro, and Lloyd, PRA 78, (2008); PRA 79, (2009); Tsang, PRL 102, (2009); PRA 80, (2009); Tsang, PRA 81, (2010). Quantum Noise Cancellation (QNC) Flowcharts QNC for Opto-mechanical Sensors Tsang and Caves, under preparation. or Google Mankei Tsang Life can only be understood backwards; but it must be lived forwards. Soren Kierkegaard Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.22/32

23 Time-Symmetric Smoothing P(x t y t0,..., y T ) = P(y future x t )P(x t y past ) R dxt (numerator) (19) Mayne, Fraser, Potter, Pardoux Unnormalized versions of P(x t δy past ) and P(δy future x t ) (retrodictive likelihood function) obey a pair of adjoint equations, one to be solved forward in time and one backward in time Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.23/32

24 Phase-Locked Loop Design Personick, IEEE Trans. Inform. Th. IT-17, 240 (1971); Wiseman, PRL 75, 4587 (1995); Armen et al., PRL 89, (2002). Berry and Wiseman, PRA 65, (2002); 73, (2006). Tsang, Shapiro, and Lloyd, PRA 78, (2008); 79, (2009); Tsang, PRA 80, (2009). Wheatley et al., PRL 104, (2010). Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.24/32

25 Phase-Locked Loop: Post-Loop Smoother j df= dt X µ (Aµf) + 1 X 2 xµ 2 µ,ν xµ xν " χ γ «(qf) + 2 q χ γ «2 p +dy t» sin(φ φ t ) 2b + p 2γq + s γ 2 q» BQB T µν f (pf)! # + γ 4 2 f q f p 2 1 ff A + cos(φ φ t ) p 2γp + s γ 2 p! f. (20) j X dg= dt Aµ µ " + χ γ «2 g xµ q g q X µ,ν BQB T µν χ γ «p g # + γ 2 p 4 +dy t» sin(φ φ t ) 2b + p 2γq s γ 2 2 g xµ xν q! 2 g q g p 2 1 ff A + cos(φ φ t ) p 2γp s γ 2 p! g. (21) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.25/32

26 Phase-Space Smoothing Convert to Wigner distributions: tr[ĝ(x, t) ˆf(x, t)] Z dqdp g(q, p, x, t)f(q, p, x, t) (22) h(x, t) = R dqdp g(q, p, x, t)f(q, p, x, t) R dx (numerator) (23) If f(q, p, x, t) and g(q, p, x, t) are non-negative, equivalent to classical smoothing: h(q, p, x, t) = g(q, p, x, t)f(q, p, x, t) R dxdqdp (numerator), h(x, t) = Z dqdp h(q, p, x, t) (24) q and p can be regarded as classical, with h(q, p, x, t) the classical smoothing probability distribution If f(q, p, x, t) and g(q, p, x, t) are Gaussian, equivalent to linear smoothing, use two Kalman filters (Mayne-Fraser-Potter smoother) Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.26/32

27 Filtering Filtering equation: d ˆf(x, t) = i dt hĥ(x), ˆf(x, t) i + dt a x + b2 2 2 x 2 «ˆf(x, t) + γ 8 dt h2ˆq ˆf(x, t)ˆq ˆq 2 ˆf(x, t) ˆf(x, t)ˆq 2 i + γ 2 dy t h ˆq ˆf(x, t) + ˆf(x, i t)ˆq, In terms of f(q, p, x, t): df = dt p m q x p + a x + b2 2 2 x γ 8 2 «p 2 f + γdy t qf. Kalman filter: dµ = Aµdt + ΣC T dσ γdη, dt = AΣ + ΣAT ΣC T γcσ + Q,! 0 0 1/m 0 A = 0 0 1, C = ( ), Q γ/ a 0 0 b 2 1 A. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.27/32

28 Steady-State Filtering µ q, µ p, Σ qq, Σ qp, and Σ pp determine conditional sensor quantum state ˆρ, Σ xx is mean-square force estimation error At steady state, let dσ/dt = 0 (solve numerically) x t = 0 limit (analytic): Σ qq = s mγ, Σ qp = 2, Σ pp = mγ 2, Σ qq Σ pp Σ 2 qp = 1 4 (25) Mensky, Continuous Quantum Measurements and Path Integrals (IOP, Bristol, 1993); Barchielli, Lanz, and Prosperi, Nuovo Cimento 72B, 79 (1982); Caves and Milburn, PRA 36, 5543 (1987); Belavkin and Staszewski, PLA 140, 359 (1989); PRA 45, 1347 (1992). Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.28/32

29 Smoothing Write retrodictive equation for ĝ(x, t) Convert ĝ(x, t) to g(q, p, x, t) solve for mean vector ν and covariance matrix Ξ of g(q, p, x, t) using a retrodictive Kalman filter Define h(q, p, x, t) = g(q, p, x, t)f(q, p, x, t) R dqdpdx g(q, p, x, t)f(q, p, x, t) (26) Mean ξ and covariance Π of h(q, p, x, t): ξ = ξ q ξ p ξ x! = Π `Σ 1 µ + Ξ 1 ν Π = Π qq Π qp Π qx Π pq Π pp Π px Π xq Π xp Π xx! = (Σ 1 + Ξ 1 ) 1 ξ x is smoothing estimate of force, Π xx is smoothing error but what are ξ q, ξ p, Π qq, Π qp, Π pp, and h(q, p, x, t) in general? Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.29/32

30 Quantum Smoothing Can we use h(q, p, x, t) = g(q, p, x, t)f(q, p, x, t) R dqdpdx (numerator) (27) to estimate q and p? when g and f are non-negative, problem becomes classical h(x, t) = Z dqdph(q, p, x, t) (28) ξ q, ξ p real part of weak values h(q, p, x, t) arises from statistics of weak measurements h(q, p, x, t) can go negative, many versions of Wigner distributions for discrete degrees of freedom Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.30/32

31 Beyond Heisenberg? Bayesian view: shouldn t we be able to learn more about a system, whether classical or quantum, in retrospect? Everett view: Need two wavefunctions or two density operators to solve problems. Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.31/32

32 Weak Measurements The smoothing estimates ξ q and ξ p are equivalent to weak values : ξ q = Re R dx tr(ĝˆq ˆf) R dx tr(ĝ ˆf), ξ p = Re R dxtr(ĝˆp ˆf) R dx tr(ĝ ˆf). (29) Aharonov, Albert, and Vaidman, PRL 60, 1351 (1988); Wiseman, PRA 65, (2002). Make weak Gaussian and backaction evading measurements of q and p: ˆM(y q ) P(y q, y p y past, y future ) Z Z dq exp " dqdp exp ǫ(y q q) 2 4 n ǫ 2 # q q, same for y p, (30) h (y q q) 2 + (y p p) 2io h(q, p, t) (31) in the limit of ǫ 0 (opposite limit to von Neumann measurements) h(q, p, t) describes the apparent statistics of q and p in the weak measurement limit Coherent Quantum Noise Cancellation for Opto-mechanical Sensors p.32/32