Reversing Time and Space in Classical and Quantum Optics

Transcription

1 Reversing Time and Space in Classical and Quantum Optics Mankei Tsang Keck Foundation Center for Extreme Quantum Information Theory, MIT Reversing Time and Space in Classical and Quantum Optics p.1/39

2 Overview Nonlinear Optics Reverse Propagation Spectral Phase Conjugation Quantum Optics Time-Entangled Photons Quantum Imaging using Path-Entangled Photons Nano-Imaging Evanescent-Wave Amplification and Conversion Quantum Sensing Bayesian Quantum Estimation Time-Symmetric Smoothing Reversing Time and Space in Classical and Quantum Optics p.2/39

3 Ultrashort Optical Pulse Propagation in Fiber Loss, Group-Velocity Dispersion, Kerr Self-Phase Modulation, Self-Steepening, Intrapulse Raman Scattering, etc. A z = α 2 A iβ A T 2 + β 3 6 3» A T 3 + iγ A 2 A + i ω 0 T ( A 2 A) T R A A 2 T Agrawal, Nonlinear Fiber Optics Phase Conjugation, Solitons, Evolution Algorithm, etc. Reversing Time and Space in Classical and Quantum Optics p.3/39

4 Reverse Propagation A z = α 2 A iβ A T 2 + β 3 6 3» A T 3 + iγ A 2 A + i ω 0 T ( A 2 A) T R A A 2 T Tsang, Psaltis, and Omenetto, Opt. Lett. 28, 1873 (2003) Barsi, Wan, and Fleischer, Nature Photonics 3, 211 (2009) Reversing Time and Space in Classical and Quantum Optics p.4/39

5 Temporal Phase Conjugation Yariv, Fekete, and Pepper, Opt. Lett. 4, 52 (1979); Fisher, Suydam, and Yevick, Opt. Lett. 8, 611 (1983). can only compensate for even-order dispersion and Kerr effect Reversing Time and Space in Classical and Quantum Optics p.5/39

6 Spectral Phase Conjugation Tsang and Psaltis, Opt. Lett. 28, 1558 (2003) Compensate for dispersion of all orders as well as Kerr effect Kuzucu et al., CLEO 2009 Postdeadline paper CPDB3 Reversing Time and Space in Classical and Quantum Optics p.6/39

7 How to Perform SPC Miller, Opt. Lett. 5, 300 (1980); Tsang and Psaltis, Optics Express 12, 2207 (2004) A (t) s x z A p (t) χ(3) d A (t) i L z = L/2 z = L/2 A q (t) Tsang and Psaltis, Opt. Commun. 242, 659 (2004) ω0 ω0 A (t) s A (t) i x χ(2) z 2ω0 A p (t) L z = L/2 z = L/2 Λ d Reversing Time and Space in Classical and Quantum Optics p.7/39

8 Spontaneous Parametric Down Conversion Ĥ χâpâ sâ i + H.c., Ψ 0 + ηa pâ sâ i 0 (1) Reversing Time and Space in Classical and Quantum Optics p.8/39

9 Spontaneous SPC A (t) i x z 2ω 0 A p(t) A (t) s χ(2) Λ d ω 0 ω 0 L z = L/2 z = L/2 A (t) i Tsang and Psaltis, Phys. Rev. A, 71, (2005) x χ(3) z A p (t) L z = L/2 z = L/2 A q (t) d A (t) s Reversing Time and Space in Classical and Quantum Optics p.9/39

10 Coincident Frequency Entanglement Giovannetti et al., Phys. Rev. Lett. 88, (2002); Kuzucu et al., Phys. Rev. Lett. 94, (2005) Reversing Time and Space in Classical and Quantum Optics p.10/39

11 SPC via Extended Phase Matching Tsang, JOSA B 23, 861 (2006) A s (L, τ) = A s (0, τ) sec G + ia i (0, τ)tan G (2) A i (L, τ) = A i (0, τ)sec G + ia s(0, τ)tan G (3) Z vχ G = 1 k p/k s dτa p (τ) (4) predicted photon pair generation rate = f R tan 2 G s 1, matches Kuzucu et al. s experiment ( s 1 ) Reversing Time and Space in Classical and Quantum Optics p.11/39

12 Mirrorless Parametric Oscillator What happens when G = π/2, tan 2 G =? Pu, Wu, Tsang, and Psaltis, Appl. Phys. Lett. 91, (2007) 43% down conversion efficiency, 140 db equivalent gain Reversing Time and Space in Classical and Quantum Optics p.12/39

13 What s Special about Coin. Freq. Entanglement? Quantum enhancement of timing accuracy [Giovannetti et al., Nature 412, 417 (2001)]. t 2 t 1 2 t 2 t 1 +t 2 2 ω 2 ω 1 +ω 2 2 ω t + ω + ω t 1 ω 1 Analogous to how mutual funds work: selecting negatively-correlated stocks reduces risk. Reversing Time and Space in Classical and Quantum Optics p.13/39

14 Multiphoton Enhancement Giovannetti, Lloyd, and Maccone, Nature 412, 417 (2001) N independent photons: T 1 2 N ω (Standard Quantum Limit) (5) e.g.: W = 100 ps, N = 10 10, T = 1 fs N negatively-time-correlated photons: T 1 2N ω (Ultimate Quantum Limit) (6) N = 2 is quite useless compared to N 1 How to create multiphoton time anti-correlation with N 1? Extended Phase Matching doesn t work when number of generated photon pairs > 1. Reversing Time and Space in Classical and Quantum Optics p.14/39

15 Adiabatic Control of Optical Solitons 1 Diffusive Jitter A(z, t) T 2 (z) D/ T 2 (0) Chirp-Induced Jitter a.u. a.u t (ps) ω (THz) 4 a(z, ω) z (m) z (m) T 2 (z) C/ T 2 (0) T 2 (z) GH/ T 2 (0) Gordon-Haus Jitter z (m) Tsang, Phys. Rev. Lett. 97, (2006); Phys. Rev. A 75, (2007) Reversing Time and Space in Classical and Quantum Optics p.15/39

16 Quantum Imaging Boto et al., Phys. Rev. Lett. 85, 2733 (2000), resolution λ/n Tsang, Phys. Rev. A 75, (2007); Phys. Rev. Lett. 101, (2008); Phys. Rev. Lett. (Editors Suggestion) 102, (2009) Reversing Time and Space in Classical and Quantum Optics p.16/39 (from wikipedia)

17 Near-Field Imaging Pendry, Phys. Rev. Lett. 85, 3966 (2000) Tsang and Psaltis, Opt. Lett. 31, 2741 (2006); Opt. Express 15, (2007); Phys. Rev. B 77, (2008) Reversing Time and Space in Classical and Quantum Optics p.17/39

18 Continuous Waveform Estimation In real-world applications, the signal x(t) to be estimated is usually a random process Nonclassical photons need to be created continuously for continuous estimation Bayesian Estimation Theory Reversing Time and Space in Classical and Quantum Optics p.18/39

19 Tracking an Aircraft or submarine, terrorist, criminal, mosquito, cancer cell, nanoparticle,... Use Bayes theorem: P(x t δy t ) = P(δy t x t )P(x t ) R dxt (numerator), (7) P(δy t x t ) from observation noise, P(x t ) from a priori information. Reversing Time and Space in Classical and Quantum Optics p.19/39

20 Prediction Assume x t is a Markov process, use Chapman-Kolmogorov equation: P(x t+δt δy t ) = Z dx t P(x t+δt x t )P(x t δy t ) (8) Reversing Time and Space in Classical and Quantum Optics p.20/39

21 Filtering: Real-Time Estimation Applying Bayes theorem and Chapman-Kolmogorov equation repeatedly, we can obtain P(x t δy t δt,..., δy t0 +δt, δy t0 ) (9) Useful for control, weather and finance forecast, etc. Wiener, Stratonovich, Kalman, Kushner, etc. Reversing Time and Space in Classical and Quantum Optics p.21/39

22 Quantum Filtering Use quantum Bayes theorem, ˆρ t ( δy t ) = ˆM(δy t )ˆρ t ˆM (δy t ) tr(numerator) (10) Use a completely positive map to evolve the system state (analogous to Chapman-Kolmogorov), ˆρ t+δt ( δy t ) = X µ ˆK µˆρ t ( δy t ) ˆK µ (11) Belavkin, Barchielli, Carmichael, Caves, Milburn, Wiseman, Mabuchi, etc. Useful for cavity QED, quantum optics, etc. Reversing Time and Space in Classical and Quantum Optics p.22/39

23 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 δy t ) = ˆM(δy t x t )ˆρ(x t ) ˆM (δy t x t ) R, (12) dxt tr(numerator) ˆρ(x t+δt δy t ) = Z dx t P(x t+δt x t ) X µ ˆK µ (x t )ˆρ(x t δy t ) ˆK µ (x t) (13) Tsang, Phys. Rev. Lett. 102, (2009); Phys. Rev. A 80, (2009). Reversing Time and Space in Classical and Quantum Optics p.23/39

24 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. (14) 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. (15) dx tr[ ˆf(x, t)] Reversing Time and Space in Classical and Quantum Optics p.24/39

25 Phase-Locked Loop Design Personick, IEEE Trans. Inform. Th. IT-17, 240 (1971); Wiseman, Phys. Rev. Lett. 75, 4587 (1995); Armen, Mabuchi et al., Phys. Rev. Lett. 89, (2002). Berry and Wiseman, Phys. Rev. A 65, (2002); 73, (2006). Tsang, Shapiro, and Lloyd, Phys. Rev. A 78, (2008); 79, (2009); Tsang, Phys. Rev. A 80, (2009). Reversing Time and Space in Classical and Quantum Optics p.25/39

26 Smoothing: Estimation with Delay More accurate than filtering Sensing, analog communication, astronomy, crime investigation,... Delay Reversing Time and Space in Classical and Quantum Optics p.26/39

27 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. Reversing Time and Space in Classical and Quantum Optics p.27/39

28 Hybrid Time-Symmetric Smoothing Use two operators to describe system: density operator ˆρ(x t δy past ) and a retrodictive likelihood operator Ê(δy future x t ) P(x t δy past, δy future ) = i tr hê(δyfuture x t )ˆρ(x t δy past ) R (16) dx(numerator) Reversing Time and Space in Classical and Quantum Optics p.28/39

29 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 δy past, δy future ) =! + 1 dy T R 1 Ĉ ĝ + ĝĉ 2 h tr ĝ(x, t) ˆf(x, i t) R dx(numerator) Tsang, Phys. Rev. Lett. 102, (2009); Phys. Rev. A 80, (2009). Classical: Pardoux, Stochastics 6, 193 (1982). Reversing Time and Space in Classical and Quantum Optics p.29/39

30 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) (17) h(x, t) = R dqdp g(q, p, x, t)f(q, p, x, t) R dx (numerator) (18) 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) (19) 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) Reversing Time and Space in Classical and Quantum Optics p.30/39

31 Phase-Locked Loop: Post-Loop Smoother Tsang, Phys. Rev. A 80, (2009). Reversing Time and Space in Classical and Quantum Optics p.31/39

32 Force Sensor Assume force is an Ornstein-Uhlenbeck process: dx t = ax t dt + bdw t (20) Hamiltonian: Ĥ(x t ) = ˆp2 2m x tˆq (21) Caves et al., RMP 52, 341 (1980); Braginsky and Khalili, Quantum Measurements (Cambridge University Press, Cambridge, 1992); Mabuchi, Phys. Rev. A 58, 123 (1998); Verstraete et al., Phys. Rev. A 64, (2001). Reversing Time and Space in Classical and Quantum Optics p.32/39

33 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. Reversing Time and Space in Classical and Quantum Optics p.33/39

34 Steady-State Filtering 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 (22) Mensky, Continuous Quantum Measurements and Path Integrals (IOP, Bristol, 1993); Barchielli, Lanz, and Prosperi, Nuovo Cimento 72B, 79 (1982); Caves and Milburn, Phys. Rev. A 36, 5543 (1987); Belavkin and Staszewski, Phys. Lett. A 140, 359 (1989); Phys. Rev. A 45, 1347 (1992). Reversing Time and Space in Classical and Quantum Optics p.34/39

35 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) (23) 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? Reversing Time and Space in Classical and Quantum Optics p.35/39

36 Filtering vs Smoothing at Steady State a/ b = 0.01/( m) 1/4, 1/β = (γ/b)( p 3 /m), 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 Position Variance s qq s qq (β = 0) π qq Momentum Variance s pp s pp (β = 0) π pp db 0 db /β /β Uncertainty Product 4(s qq s pp -s 2 qp) Heisenberg 4π qq π pp Signal-to-Noise Ratio filtering smoothing db db /β /β Reversing Time and Space in Classical and Quantum Optics p.36/39

37 Quantum Smoothing Can we use h(q, p, x, t) = g(q, p, x, t)f(q, p, x, t) R dqdpdx (numerator) (24) to estimate q and p? when g and f are non-negative, problem becomes classical h(x, t) = Z dqdph(q, p, x, t) (25) ξ q, ξ p real part of weak values arises from statistics of weak measurements h(q, p, x, t) can go negative, many versions of Wigner distributions for discrete degrees of freedom Reversing Time and Space in Classical and Quantum Optics p.37/39

38 Sensing Beyond Heisenberg? An engineer would conclude that the sensor did have sub-heisenberg uncertainties some time in the past to enable the enhanced sensing accuracy Shouldn t we be able to learn more about a system, whether classical or quantum, in retrospect? Reversing Time and Space in Classical and Quantum Optics p.38/39

39 Summary Reverse Propagation and Spectral Phase Conjugation: Tsang, Psaltis, and Omenetto, Opt. Lett. 28, 1873 (2003); Tsang and Psaltis, Opt. Lett. 28, 1558 (2003); Optics Express 12, 2207 (2004); Opt. Commun. 242, 659 (2004). A (t) s x χ(3) z A p (t) d Generation of Temporally Anti-correlated Entangled Photons: Tsang and Psaltis, Phys. Rev. A 71, (2005); 73, (2006); Tsang, JOSA B 23, 861 (2006); Phys. Rev. Lett. 97, (2006); Phys. Rev. A 75, (2007); Pu, Wu, Tsang, and Psaltis, Appl. Phys. Lett. 91, (2007). A (t) i L z = L/2 z = L/2 A q (t) Quantum Imaging and Near-Field Imaging: Tsang, Phys. Rev. A 75, (2007); Phys. Rev. Lett. 101, (2008); Phys. Rev. Lett. (Editors Suggestion) 102, (2009); Tsang and Psaltis, Opt. Lett. 31, 2741 (2006); Opt. Express 15, (2007); Phys. Rev. B 77, (2008) Quantum Smoothing: Tsang, Shapiro, and Lloyd, Phys. Rev. A 78, (2008); 79, (2009); Tsang, Phys. Rev. Lett. 102, (2009); Phys. Rev. A 80, (2009); e-print arxiv: mankei@mit.edu Reversing Time and Space in Classical and Quantum Optics p.39/39