General Theory of Quantum Sensors: Estimation, Control ... · General Theory of Quantum Sensors:...
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General Theory of Quantum Sensors:Estimation, Control, and Fundamental Limits
Mankei Tsang
Center for Quantum Information and Control, UNM
Keck Foundation Center for Extreme Quantum Information Theory, MIT
Department of Electrical Engineering, Caltech
General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.1/19
Quantum Systems for Sensing
10dB squeezing, Vahlbruch etal., PRL 100, 033602 (2008).
Julsgaard, Kozhekin, andPolzik, Nature 413, 400 (2001).
Rugar et al., Nature 430, 329 (2004).
Neeley et al., Nature 467, 570 (2010)
O’Connell et al., Nature 464, 697 (2010).
Kippenberg and Vahala, Science 321, 1172 (2008), and ref-erences therein.
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Outline
Quantum EstimationM. Tsang, J. H. Shapiro, and S. Lloyd,Phys. Rev. A 78, 053820 (2008); 79, 053843(2009).
M. Tsang, “Time-Symmetric Quantum Theory ofSmoothing,” Phys. Rev. Lett. 102, 250403 (2009).
M. Tsang, Phys. Rev. A 80, 033840 (2009); 81,013824 (2010).
Quantum Noise ControlM. Tsang and C. M. Caves, “CoherentQuantum-Noise Cancellation for OptomechanicalSensors,” Phys. Rev. Lett. 105, 123601 (2010).
Fundamental Quantum LimitM. Tsang, H. M. Wiseman, and C. M. Caves,“Fundamental Quantum Limit to WaveformEstimation,” e-print arXiv:1006.5407.
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Other Topics
Quantum Imaging
M. Tsang, “Quantum Imaging beyond the DiffractionLimit by Optical Centroid Measurements,”Phys. Rev. Lett. (Editors’ Suggestion) 102, 253601(2009).
M. Tsang, Phys. Rev. Lett. 101, 033602 (2008).
M. Tsang, Phys. Rev. A 75, 043813 (2007).
Quantum OpticsM. Tsang, Phys. Rev. A 81, 063837 (2010).
M. Tsang, Phys. Rev. Lett. 97, 023902 (2006).
M. Tsang, Phys. Rev. A 75, 063809 (2007).
M. Tsang and D. Psaltis, Phys. Rev. A 73, 013822(2006).
M. Tsang and D. Psaltis, Phys. Rev. A 71, 043806(2005).
Superresolution Imaging
M. Tsang and D. Psaltis, “Magnifying perfect lens andsuperlens design by coordinate transformation,”Phys. Rev. B 77, 035122 (2008).
M. Tsang and D. Psaltis, Optics Express 15, 11959(2007).
M. Tsang and D. Psaltis, Optics Lett. 31, 2741 (2006).
Ultrafast Nonlinear OpticsY. Pu, J. Wu, M. Tsang, and D. Psaltis,Appl. Phys. Lett. 91, 131120 (2007).
M. Tsang, J. Opt. Soc. Am. B 23, 861 (2006).
M. Centurion, Y. Pu, M. Tsang, and D. Psaltis,Phys. Rev. A 71, 063811 (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).
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Basic Problem of Quantum Estimation
Given observation record Yτ ≡ {yt; t0 ≤ t < τ}, what is xt (e.g. classical force,gravitational wave, magnetic field)?
Bayesian approach: calculate or approximate P (xt|Yτ )
Classical Bayesian Estimation: Radar, aircraft control, robotics, remote sensing, GPS,astronomy, bio-imaging, weather forecast, finance, credit card fraud detection, crimeinvestigation, . . .
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Quantum Smoothing
M. Tsang, “Time-Symmetric Quantum Theory of Smoothing,” Phys. Rev. Lett. 102,250403 (2009).
df = dtL(x)f +dt
8
2CT
R−1
fC†
− C†T
R−1
Cf − fC†T
R−1
C
!
+1
2dy
Tt R
−1“
Cf + fC†”
−dg = dtL∗(x)g +
dt
8
2C†T
R−1
gC − C†T
R−1
Cg − gC†T
R−1
C
!
+1
2dy
Tt R
−1“
C†
g + gC”
P (xt = x|Ypast, Yfuture) =trh
g(x, t)f(x, t)i
R
dx(numerator)
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Adaptive Quantum Optical Phase Estimation
Personick, IEEE Trans. Inform. Th. IT-17, 240 (1971); Wiseman, PRL 75, 4587 (1995);Armen et al., PRL 89, 133602 (2002); Berry and Wiseman, Phys. Rev. A 65, 043803(2002); 73, 063824 (2006).
M. Tsang, J. H. Shapiro, and S. Lloyd, Phys. Rev. A 78, 053820 (2008); 79, 053843(2009).
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Experimental Demonstration
Wheatley et al., “Adaptive Optical Phase Estimation Using Time-Symmetric QuantumSmoothing,” Phys. Rev. Lett. (Editors’ Suggestion) 104, 093601 (2010).
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Optomechanical Force Sensor
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Noise Cancellation
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Broadband Quantum Noise Cancellation
M. Tsang and C. M. Caves,Phys. Rev. Lett. 105, 123601(2010).
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Fundamental Quantum Limit
Quantum Cramér-Rao Bound:
〈δx2〉t≥ F−1(t, t), (1)
Z
dt′F (t, t
′)F
−1(t
′, τ) = δ(t − τ), F = F
(Q)+ F
(C), (2)
F(Q)
(t, t′) =
4
~2
D
∆h(t)∆h(t′)E
, h(t) ≡
Z
tJ
t0
dτU†(τ, t0)
δH(τ)
δx(t)U(τ, t0), (3)
F(C)
(t, t′) =
Z
DxP [x]δ ln P [x]
δx(t)
δ ln P [x]
δx(t′). (4)
M. Tsang, H. M. Wiseman, and C. M. Caves, “Fundamental Quantum Limit toWaveform Estimation,” e-print arXiv:1006.5407.
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Optimal Optomechanical Force Sensing
Optimal Estimation (Smoothing) + Noise Control (QNC) saturate the FundamentalQuantum Limit (Quantum Cramér-Rao Bound).
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Extensions and Generalizations
Quantum Generalizations of Detection, Estimation, and Modulation Theory
Van Trees, Detection, Estimation, and Modulation Theory
Array Signal Processing, Imaging
Nonlinear, Non-Gaussian Estimation Techniques
Novel Signal Processing: e.g. Compressive Sensing
Quantum Noise Control
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Applications
Atomic Magnetometry
Julsgaard, Kozhekin, andPolzik, Nature 413, 400 (2001).
Rugar et al., Nature 430, 329 (2004).
Wildermuth et al., Nature 435, 440(2005).
Opto- and Electro-Mechanical Force Sensing
Kippenberg and Vahala, Science 321, 1172 (2008), and ref-erences therein.
Optical Interferometry and Imaging
Tsang, Phys. Rev. Lett. (Editors’ Suggestion)102, 253601 (2009)
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Collaborations with Experimentalists
Elanor Huntington’s group at ADFA@UNSW, Canberra, Australia on adaptive opticalphase estimation
Michele Heurs’s group at Albert Einstein Institute at Hannover, Germany on quantumnoise cancellation for optomechanics
DARPA proposal with Dima Budker at Berkeley, Louis Bouchard and Kang Wang atUCLA, Phillip Hemmer at Texas A&M, Zac Dutton at BBN ondiamond-nitrogen-vacancy-center magnetometry
George Barbastathis’s group at MIT and SMART on imaging
Waller et al., to appear in Optics Express
More collaborators (both quantum and classical) are welcome.
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Future of Science and Engineering
Quantum sensors are now reality
Opto- and electro-mechanical force sensorAtomic magnetometry
Optical interferometry
Optical imaging
Future science and technology will require increasingly precise knowledge and controlof space, time, and energy.
Nanotechnology and beyond
“Battle-tested” engineering methodologies
Bayesian estimation
Control theory
Engineering tools will benefit fundamental physics as well.
Metrology
Foundations
Will quantum effects help classical applications?
Quantum superresolution imaging
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Singapore
Quantum information: critical mass
Engineer’s perspective
Collaborations with researchers on sensing, imaging, quantum optics, nonlinear optics,nano-optics, . . .
Quantum Imaging
Metamaterials, Nano-Optics
Tsang, Phys. Rev. B 77, 035122 (2008).
Quantum Optics, Nonlinear Optics
Tsang, Phys. Rev. A 81, 063837 (2010).
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Bayesian Estimation
Bayesian estimation: calculate conditional probability P (xt|Yτ ) using measurementrecord Yτ ≡ {y(t); t0 ≤ t < τ},
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
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