General Theory of Quantum Sensors:Estimation, Control, and Fundamental Limits

Mankei Tsang

mankei@unm.edu

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.

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.2/19

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.

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.3/19

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).

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.4/19

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, . . .

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.5/19

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)

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.6/19

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).

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.7/19

Experimental Demonstration

Wheatley et al., “Adaptive Optical Phase Estimation Using Time-Symmetric QuantumSmoothing,” Phys. Rev. Lett. (Editors’ Suggestion) 104, 093601 (2010).

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.8/19

Optomechanical Force Sensor

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.9/19

Noise Cancellation

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.10/19

Broadband Quantum Noise Cancellation

M. Tsang and C. M. Caves,Phys. Rev. Lett. 105, 123601(2010).

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.11/19

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.

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.12/19

Optimal Optomechanical Force Sensing

Optimal Estimation (Smoothing) + Noise Control (QNC) saturate the FundamentalQuantum Limit (Quantum Cramér-Rao Bound).

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.13/19

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

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.14/19

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)

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.15/19

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.

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.16/19

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

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.17/19

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).

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.18/19

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

General Theory of Quantum Sensors: Estimation, Control, and Fundamental Limits – p.19/19