Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control

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1 Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control Mankei Tsang Center for Quantum Information and Control, UNM Keck Foundation Center for Extreme Quantum Information Theory, MIT Department of Electrical Engineering, Caltech Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.1/26

2 Quantum Systems for Sensing 10dB squeezing, Vahlbruch et al., PRL 100, (2008). Neeley et al., Nature 467, 570 (2010) Julsgaard, Kozhekin, and Polzik, Nature 413, 400 (2001). Rugar et al., Nature 430, 329 (2004). O Connell et al., Nature 464, 697 (2010). Kippenberg and Vahala, Science 321, 1172 (2008), and references therein. Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.2/26

3 Outline Fundamental Quantum Limit M. Tsang, H. M. Wiseman, and C. M. Caves, Fundamental Quantum Limit to Waveform Estimation, e-print arxiv: Quantum Estimation M. Tsang, J. H. Shapiro, and S. Lloyd, Phys. Rev. A 78, (2008); 79, (2009). M. Tsang, Time-Symmetric Quantum Theory of Smoothing, Phys. Rev. Lett. 102, (2009). M. Tsang, Phys. Rev. A 80, (2009); 81, (2010). Seminar next Monday Quantum Noise Control M. Tsang and C. M. Caves, Coherent Quantum-Noise Cancellation for Optomechanical Sensors, Phys. Rev. Lett. 105, (2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.3/26

4 Other Topics Superresolution Imaging Quantum Imaging M. Tsang, Quantum Imaging beyond the Diffraction Limit by Optical Centroid Measurements, Phys. Rev. Lett. (Editors Suggestion) 102, (2009). M. Tsang, Phys. Rev. Lett. 101, (2008). M. Tsang, Phys. Rev. A 75, (2007). Quantum Optics M. Tsang, Phys. Rev. A 81, (2010). M. Tsang, Phys. Rev. Lett. 97, (2006). M. Tsang, Phys. Rev. A 75, (2007). M. Tsang and D. Psaltis, Phys. Rev. A 73, (2006). M. Tsang and D. Psaltis, Phys. Rev. A 71, (2005). M. Tsang and D. Psaltis, Magnifying perfect lens and superlens design by coordinate transformation, Phys. Rev. B 77, (2008). M. Tsang and D. Psaltis, Optics Express 15, (2007). M. Tsang and D. Psaltis, Optics Lett. 31, 2741 (2006). Ultrafast Nonlinear Optics Y. Pu, J. Wu, M. Tsang, and D. Psaltis, Appl. Phys. Lett. 91, (2007). M. Tsang, J. Opt. Soc. Am. B 23, 861 (2006). M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, Phys. Rev. A 71, (2005). M. Tsang and D. Psaltis, Opt. Commun. 242, 659 (2004). M. Tsang and D. Psaltis, Opt. Express 12, 2207 (2004). M. Tsang, D. Psaltis, and F. G. Omenetto, Opt. Lett. 28, 1873 (2003). M. Tsang and D. Psaltis, Opt. Lett. 28, 1558 (2003). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.4/26

5 Heisenberg Uncertainty Principle ˆq 2 ˆp 2 t 2 t 4. (1) Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.5/26

6 Quantum Limit to Sensing LIGO, Livingston, Louisiana Sensor (mechanical, atomic spin ensembles) is quantum, but the signal of interest (force, magnetic field) is classical. Quantum limit to sensing due to Heisenberg principle: Braginsky and Khalili, Quantum Measurements (Cambridge University Press, Cambridge, 1992); C. M. Caves et al., Rev. Mod. Phys. 52, 341 (1980); H. P. Yuen, PRL 51, 719 (1983),... Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.6/26

7 Parameter-Based Uncertainty Relation phase modulation by unknown classical parameter x: ˆρ x = exp( iĥx)ˆρ 0 exp(iĥx) i Measurement probability density = P(y x) = tr hê(y)ˆρx Use y to estimate x, estimate = x(y). Mean-square error δx 2 R dy ( x x) 2 P(y x). Quantum Cramér-Rao bound (QCRB): δx 2 D ĥ2e 2 4. (2) C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976); V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.7/26

8 Problem 1: x(t) Changes in Time Realizations of Wiener Process Gravitational waves may be stochastic [Buonanno et al., Phys. Rev. D 55, 3330 (1997); Apreda et al., Nuclear Physics B 631, 342 (2002)]. x(t) can be a vector. Stochastic process characterized by a priori probability density functional P[x(t)] Popular assumptions: Gaussian: P exp[ R dtdt x(t)k 1 (t, t )x(t )] Stationary: x(t)x(t ) = K(t t), S x (ω) = R dτk(τ)exp(iωτ) Markovian: P[x(t)] = ˆQ n=1 P(x n+1 x n ) P(x 0 ) Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.8/26

9 Problem 2: Continuous Dynamics and Measurements Continuous coupling of time-varying signal to sensor Continuous quantum dynamics of sensor Continuous non-destructive measurements: must include measurement back-action Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.9/26

10 Quantum Information Theory to the Rescue Discretize time Measurements and dynamics described by a sequence of completely positive maps represented by Kraus operators P[y x] = X µ N,...,µ 1 tr h ˆKµN (y N x N )... ˆK µ1 (y 1 x 1 )ρ 0 ˆK µ1 (y 1 x 1 )... ˆK µn (y N x N )i (3) Equivalent quantum circuit model (Kraus representation theorem, principle of deferred measurements): K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory (Springer, Berlin, 1983). M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.10/26

11 Dynamical Quantum Cramér-Rao Bound Fisher information matrix F(t, t ) Define inverse of F(t, t ) as R dt F(t, t )F 1 (t, τ) = δ(t τ). δx 2 t F 1 (t, t). (4) Two components: F(t, t ) = F (Q) (t, t ) + F (C) (t, t ). F (Q) is a two-time quantum covariance function: F (Q) (t, t ) = 4 D ĥ(t) ĥ(t E 2 ), ĥ(t) Z tj t 0 dτû (τ, t 0 ) δĥ(τ) Û(τ, t 0 ). δx(t) F (C) incorporates a priori waveform information F (C) (t, t ) = Z δ ln P[x] DxP[x] δx(t) δ ln P[x] δx(t ). (5) M. Tsang, H. M. Wiseman, and C. M. Caves, Fundamental Quantum Limit to Waveform Estimation, e-print arxiv: Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.11/26

12 Example: Optomechanical Force Sensing Assume Gaussian stationary statistics, S ˆq (ω) R dτ ˆq(t) ˆq(t + τ) exp(iωτ) δf 2 Z dω 2π 1 (4/ 2 )S ˆq (ω) + 1/S f (ω) (6) How to saturate the bound? Bayesian Quantum Estimation Classical and Quantum Noise Control Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.12/26

13 Bayesian Estimation Bayesian estimation: calculate conditional probability P(x t Y τ ) given measurement record Y τ {y(t); t 0 t < τ}. Bayes theorem: P(x y) = P(y x)p(x)/[ R dxp(y x)p(x)] Radar, aircraft control, robotics, remote sensing, GPS, astronomy, bio-imaging, weather forecast, finance, credit card fraud detection, crime investigation,... Filtering, Prediction: real-time or advanced estimation Smoothing: delayed estimation, most accurate when x(t) is a stochastic process Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.13/26

14 Smoothing for Quantum Sensing M. Tsang, Time-Symmetric Quantum Theory of Smoothing, Phys. Rev. Lett. 102, (2009); Phys. Rev. A 80, (2009); 81, (2010). Seminar next Monday Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.14/26

15 Adaptive Optical Phase Estimation Personick, IEEE Trans. Inform. Th. IT-17, 240 (1971); Wiseman, PRL 75, 4587 (1995); Armen et al., PRL 89, (2002); Berry and Wiseman, Phys. Rev. A 65, (2002); 73, (2006); M. Tsang, J. H. Shapiro, and S. Lloyd, Phys. Rev. A 78, (2008); 79, (2009). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.15/26

16 Experimental Demonstration Wheatley et al., Adaptive Optical Phase Estimation Using Time-Symmetric Quantum Smoothing, Phys. Rev. Lett. (Editors Suggestion) 104, (2010). very close to QCRB lower QCRB by optical squeezing Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.16/26

17 Optomechanical Force Sensor y(ω) = η 2 (ω) = G(ω)[f(ω) + z(ω)], S z (ω) = total noise power spectral density = Z δf 2 smoothing = dω 1 2π 1/S z (ω) + 1/S f (ω) 1 G(ω) 2 S 2(ω) + S 1 (ω) > δf 2 QCRB = Z dω 2π G(ω) 1 (4/ 2 )S ˆq (ω) + 1/S f (ω). S 2 (ω): noise in optical phase quadrature, S 1 (ω) radiation-pressure fluctuations If we can remove S 1 (ω), δf 2 smoothing can saturate QCRB Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.17/26

18 Noise Cancellation Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.18/26

19 Broadband Quantum Noise Cancellation Negative-mass ponderomotive squeezing Coherent Feedforward Quantum Control M. Tsang and C. M. Caves, Phys. Rev. Lett. 105, (2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.19/26

20 Optimal Force Sensing S z (ω) = 1 G(ω) 2 S 2 2(ω) 4 G(ω 2 S 1 (ω) = 2 4S ˆq (ω), (7) δf δf 2 2 smoothing QCRB (8) Optical squeezing of S 2 (ω) can lower the QCRB. Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.20/26

21 Summary Quantum Limit: Dynamical QCRB M. Tsang, H. M. Wiseman, and C. M. Caves, Fundamental Quantum Limit to Waveform Estimation, e-print arxiv: Estimation: Quantum Smoothing M. Tsang, J. H. Shapiro, and S. Lloyd, Phys. Rev. A 78, (2008); 79, (2009). M. Tsang, Time-Symmetric Quantum Theory of Smoothing, Phys. Rev. Lett. 102, (2009). M. Tsang, Phys. Rev. A 80, (2009); 81, (2010). Quantum Noise Control: Quantum Noise Cancellation M. Tsang and C. M. Caves, Coherent Quantum-Noise Cancellation for Optomechanical Sensors, Phys. Rev. Lett. 105, (2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.21/26

22 Future Work Publications Magnetometry Imaging DARPA proposal with Berkeley, UCLA, Texas A&M, BBN on diamond-nitrogen-vacancy-center magnetometry How low can QCRB get? Quantum smoothing for more complex systems Collaboration with Elanor Huntington s group at ADFA@UNSW in Canberra, Australia Quantum noise cancellation Collaborations with Elanor s group, Michele Heur s group at Albert Einstein Institute at Hannover, Germany Generalizations, applications to other systems Other topics: Quantum optics, nonlinear optics, nano-optics? Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.22/26

23 Quantum Smoothing? Quantum theory is a predictive theory Quantum state described by Ψ t or ˆρ t can only be conditioned only upon past observations and evolves only forward in time using Schrödinger equation or master equation Quantum filtering: Belavkin, Barchielli, Carmichael,... Smoothing cannot be applied to quantum systems: Belavkin, Foundation of Physics 24, 685 (1994). Weak values by Aharonov, Vaidman, et al. Paradox: e.g. L. Hardy, Phys. Rev. Lett. 68, 2981 (1992); Aharonov et al., Phys. Lett. A 301, 130 (2002); M. Tsang, Phys. Rev. A 81, (2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.23/26

24 Kimble Filters W. G. Unruh, in Quantum Optics, Experimental Gravitation, and Measurement Theory, edited by P. Meystre and M. O. Scully (Plenum, New York, 1982), p. 647; Kimble et al., PRD 65, (2001); Mow-Lowry et al., PRL 92, (2004). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.24/26

25 Input/Output Noise Cancellation M. Tsang and C. M. Caves, Phys. Rev. Lett. 105, (2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.25/26

26 Quantum Noise Cancellation for Magnetometry Julsgaard, Kozhekin, and Polzik, Experimental long-lived entanglement of two macroscopic objects, Nature 413, 400 (2001). Wasilewski et al., Quantum Noise Limited and Entanglement-Assisted Magnetometry, Phys. Rev. Lett. 104, (2010). Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control p.26/26