theory of quantum sensing: fundamental limits, estimation, and …€¦ · theory of quantum...

Theory of Quantum Sensing: FundamentalLimits, Estimation, and Control

Mankei Tsang

[email protected]

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

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.


Outline

Fundamental Quantum LimitM. Tsang , H. M. Wiseman, and C. M. Caves,“Fundamental Quantum Limit to WaveformEstimation,” e-print arXiv:1006.5407.

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).

Seminar next Monday

Quantum Noise ControlM. Tsang and C. M. Caves, “CoherentQuantum-Noise Cancellation for OptomechanicalSensors,” Phys. Rev. Lett. 105, 123601 (2010).


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).


Heisenberg Uncertainty Principle

˙

∆q2¸

t

˙

∆p2¸

t≥

~2

4. (1)


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, QuantumMeasurements (Cambridge University Press, Cambridge, 1992); C. M. Caves et al.,Rev. Mod. Phys. 52, 341 (1980); H. P. Yuen, PRL 51, 719 (1983), . . .


Parameter-Based Uncertainty Relation

phase modulation by unknown classical parameter x: ρx = exp(−ihx)ρ0 exp(ihx)

Measurement probability density = P (y|x) = trh

E(y)ρx

i

Use y to estimate x, estimate = x(y). Mean-square error˙

δx2¸

≡R

dy (x − x)2 P (y|x).

Quantum Cramér-Rao bound (QCRB):

˙

δx2¸

D

∆h2E

≥~2

4. (2)

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, NewYork, 1976); V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).


Problem 1: x(t) Changes in Time

0 100 200 300 400 500 600 700 800 900 1000−80

−60

−40

−20

0

20

40

60

5 Realizations of WienerProcess

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), Sx(ω) =R ∞

−∞dτK(τ) exp(iωτ)

Markovian: P [x(t)] =ˆQ

n=1 P (xn+1|xn)˜

P (x0)Theory of Quantum Sensing: Fundamental Limits, Estimation, and Control – p.8/26

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


Quantum Information Theory to the Rescue

Discretize time

Measurements and dynamics described by a sequence of completely positive mapsrepresented by Kraus operators

P [y|x] =X

µN ,...,µ1

trh

KµN(yN |xN ) . . . Kµ1

(y1|x1)ρ0K†µ1

(y1|x1) . . . K†µN

(yN |xN )i

(3)

Equivalent quantum circuit model (Kraus representation theorem, principle of deferredmeasurements):

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).


Dynamical Quantum Cramér-Rao Bound

Fisher information matrix F (t, t′)

Define inverse of F (t, t′) asR

dt′F (t, t′)F−1(t′, τ) = δ(t − τ).

〈δx2〉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

~2

D

∆h(t)∆h(t′)E

, h(t) ≡

Z tJ

t0

dτU†(τ, t0)δH(τ)

δx(t)U(τ, t0).

F (C) incorporates a priori waveform information

F (C)(t, t′) =

Z

DxP [x]δ ln P [x]

δx(t)

δ ln P [x]

δx(t′). (5)

M. Tsang , H. M. Wiseman, and C. M. Caves, “Fundamental Quantum Limit toWaveform Estimation,” e-print arXiv:1006.5407.


Example: Optomechanical Force Sensing

Assume Gaussian stationary statistics, S∆q(ω) ≡R ∞

−∞dτ 〈∆q(t)∆q(t + τ)〉 exp(iωτ)

˙

δf2¸

≥

Z ∞

−∞

dω

2π

1

(4/~2)S∆q(ω) + 1/Sf (ω)(6)

How to saturate the bound?Bayesian Quantum Estimation

Classical and Quantum Noise Control


Bayesian Estimation

Bayesian estimation: calculate conditional probability P (xt|Yτ ) given measurementrecord Yτ ≡ {y(t); t0 ≤ 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


Smoothing for Quantum Sensing

M. Tsang , “Time-Symmetric Quantum Theory of Smoothing,” Phys. Rev. Lett. 102,250403 (2009); Phys. Rev. A 80, 033840 (2009); 81, 013824 (2010).

Seminar next Monday


Adaptive 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 andWiseman, 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).


Experimental Demonstration

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

very close to QCRB

lower QCRB by optical squeezingTheory of Quantum Sensing: Fundamental Limits, Estimation, and Control – p.16/26

Optomechanical Force Sensor

y(ω) = η2(ω) = G(ω)[f(ω) + z(ω)],

Sz(ω) = total noise power spectral density =1

|G(ω)|2S2(ω) + S1(ω) ≥

~

|G(ω)|

˙

δf2¸

smoothing=

Z ∞

−∞

dω

2π

1

1/Sz(ω) + 1/Sf (ω)>

˙

δf2¸

QCRB=

Z ∞

−∞

dω

2π

1

(4/~2)S∆q(ω) + 1/Sf (ω).

S2(ω): noise in optical phase quadrature, S1(ω) radiation-pressure fluctuations

If we can remove S1(ω), 〈δf2〉smoothing can saturate QCRB


Noise Cancellation


Broadband Quantum Noise Cancellation

Negative-mass ponderomotive squeezing

Coherent Feedforward Quantum Control

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


Optimal Force Sensing

Sz(ω) =1

|G(ω)|2S2(ω) ≥

~2

4|G(ω|2S1(ω)=

~2

4S∆q(ω), (7)

˙

δf2¸

smoothing≥

˙

δf2¸

QCRB(8)

Optical squeezing of S2(ω) can lower the QCRB.


Summary

Quantum Limit: Dynamical QCRB

M. Tsang , H. M. Wiseman, and C. M. Caves,“Fundamental Quantum Limit to WaveformEstimation,” e-print arXiv:1006.5407.

Estimation: Quantum Smoothing

M. 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 Control: Quantum Noise CancellationM. Tsang and C. M. Caves, “CoherentQuantum-Noise Cancellation for OptomechanicalSensors,” Phys. Rev. Lett. 105, 123601 (2010).


Future Work

Publications

Magnetometry

DARPA proposal with Berkeley, UCLA, Texas A&M, BBN ondiamond-nitrogen-vacancy-center magnetometry

Imaging

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 cancellationCollaborations with Elanor’s group, Michele Heur’s group at Albert EinsteinInstitute at Hannover, Germany

Generalizations, applications to other systems

Other topics: Quantum optics, nonlinear optics, nano-optics

?


Quantum Smoothing?

Quantum theory is a predictive theory

Quantum state described by |Ψt〉 or ρt can only beconditioned only upon past observations and evolves onlyforward in time using Schrödinger equation or masterequation

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, 013824 (2010).


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, 022002 (2001); Mow-Lowry et al., PRL 92, 161102 (2004).


Input/Output Noise Cancellation

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


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, 133601 (2010).


theory of quantum sensing: fundamental limits, estimation, and …€¦ · theory of quantum...

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