Theory of Quantum Sensing

Fundamental Limits, Estimation, and Control

Mankei Tsangeletmk@nus.edu.sg

http://mankei.tsang.googlepages.com/

ECE, Physics, NUS

Center for Quantum Information and Control, UNM

Keck Foundation Center for Extreme Quantum Information Theory, MIT

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Quantum Sensing

Fundamental Limit: What is the ultimate sensitivity allowed by quantum mechanics?

Estimation: Optimize data processing

Control: Optimize experiment

Examples: optical interferometry, optical imaging, optomechanical force sensing(gravitational-wave detection), atomic magnetometry, electrometer, etc.

Cavity quantum electro-optics

Quantum nonlocality in stellar interferometry Theory of Quantum Sensing Fundamental Limits, Estimation, and Control – p. 2/25

Experimental Advances

10dB squeezing, Vahlbruchet al., Phys. Rev. Lett.100, 033602 (2008).

Julsgaard, Kozhekin,and Polzik, Nature413, 400 (2001).

Rugar et al., Nature 430, 329(2004).

Kimble, Nature 453, 1023 (2008).

Chou et al., Science 329, 1630(2010).

Castellanos-Beltran et al.,Nature Phys. 4, 928 (2008).

Neeley et al., Na-ture 467, 570 (2010)

O’Connell et al., Nature 464,697 (2010).

Kippenberg and Vahala, Science 321, 1172(2008), and references therein.

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Heisenberg Uncertainty Principle

Heisenberg uncertainty principle:˙

∆q2¸˙

∆p2¸

≥ ~2/4.

Sensor (mechanical, atomic spin ensembles) is quantum, but signal of interest (force,magnetic field) is classical.

Standard 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),. . .

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Parameter-Based Uncertainty Relation

Quantum Cramér-Rao bound (QCRB):

˙

δx2¸

D

∆h2E

≥~2

4. (1)

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, NewYork, 1976); V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004);Nature Photon. 5, 222 (2011).

“Heisenberg” limit (Margolus-Levitin bound): GLM, e-print arXiv:1109.5661

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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

Realizations of Wiener Process

Optical phase, magnetic field, electric field, force, etc. can all vary randomly in time

Gravitational waves are stochastic [Buonanno et al., Phys. Rev. D 55, 3330 (1997);Apreda et al., Nuclear Physics B 631, 342 (2002)].

Stochastic processes characterized by a priori statistics (e.g. Langevin,Fokker-Planck, power spectral density)

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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 backaction

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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

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).

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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). (2)

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′). (3)

M. Tsang, H. M. Wiseman, and C. M. Caves, PRL 106, 090401 (2011).

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Example 1: Adaptive Optical Phase Estimation

〈δφ2〉QCRB =

Z ∞

−∞

dω

2π

1

4S∆I(ω) + 1/Sφ(ω)

, (4)

e.g. Scoh∆I

(ω) =P

~ω0, SOU

φ (ω) =κ

ω2 + ǫ2. (5)

Filtering (quantum trajectory): Berry and Wiseman, PRA 65, 043803 (2002); 73, 063824 (2006).

Filtering does not saturate QCRB!

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Classical Estimation

Filtering, Prediction: real-time or advanced estimation

Smoothing: delayed estimation, more accurate when x(t) is a stochastic process

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Smoothing for Quantum Sensing

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

Weak Value is smoothing estimation of quantum observables, does not always makesense: Aharonov et al., PRL 60, 1351 (1988); Aharonov et al., Phys. Lett. A 301, 130(2002); Yanagisawa, e-print arXiv:0711.3885; M. Tsang, PRA 81, 013824 (2010).

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Experimental Demonstration

Optical phase-locked loop with smoother: M. Tsang, J. H. Shapiro, and S. Lloyd, PRA78, 053820 (2008); 79, 053843 (2009); M. Tsang, PRA 80, 033840 (2009):

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

very close to QCRB for coherent state.

QCRB can be lowered by squeezing Theory of Quantum Sensing Fundamental Limits, Estimation, and Control – p. 13/25

Example 2: Optomechanical Force Sensing

˙

δf2¸

QCRB=

Z ∞

−∞

dω

2π

1

(4/~2)S∆q(ω) + 1/Sf (ω). (6)

Smoothing can’t saturate QCRB due to the presence of measurement backactionnoise

Standard Quantum Limit (Braginsky, Caves et al.): backaction noise cannot beremoved.

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Quantum Noise Cancellation (QNC)

Coherent Feedforward Quantum Control

M. Tsang and C. M. Caves, PRL 105, 123601 (2010).

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Noise Cancellation

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Optimal Force Sensing

Smoothing + QNC saturate QCRB for coherent state.

Optical squeezing of input light can lower the QCRB.

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Example 3: Magnetometry

Smoothing for magnetometry with linear Gaussian approximation: Petersen andMølmer, PRA 74, 043802 (2006).

QNC:Julsgaard, Kozhekin, and Polzik, “Experimental long-lived entanglement of twomacroscopic objects,” Nature 413, 400 (2001).

Wasilewski et al., “Quantum Noise Limited and Entanglement-AssistedMagnetometry,” PRL 104, 133601 (2010).

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Measuring Quantum Systems

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Cavity Quantum Electro-Optics

M. Tsang, PRA 81, 063837 (2010); e-printarXiv:1105.2336 (2011) [accepted by PRA].

Electro-optic (Pockels) effect: applied voltagechanges optical refractive index

Physics exactly the same as cavityoptomechanics

Three-wave mixing (vs four-wave mixing inelectro-optomechanics)

Laser cooling of microwave resonator

Convert between microwave photons and opticalphotons

Electro-optic parametric amplification,entanglement

Optical QND measurements of microwave fields

Classical LiNbO3 device demonstrations byIlchenko et al., JOSA B 20, 333 (2003), etc.

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Stellar Interferometry

Gottesman, Jennewein, and Croke, e-printarXiv:1107.2939.

Estimation of coherence:

Γab = 〈b†a〉, g =〈b†a〉

p

〈b†b〉〈a†a〉(normalized). (7)

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Old-School Quantum Optics

P representation:

ρ =

Z

d2αd2βΦ(α, β)|α, β〉〈α, β|. (8)

Φ(α, β) is a two-mode zero-mean Gaussian for thermal light, i.e. no entanglement

weak thermal light ǫ ≡ 〈a†a〉 = 〈b†b〉 ≪ 1 in photon-number basis:

ρ = (1 − ǫ)|0, 0〉〈0, 0| +ǫ

2[|0, 1〉〈1, 0| + |1, 0〉〈1, 0| + g∗|0, 1〉〈1, 0| + g|1, 0〉〈0, 1|]

+ O(ǫ2), (9)

P (y|g) = tr [E(y)ρ] . (10)

Classical Fisher information for g = g1 + ig2:

Fjk =

fi

∂

∂gj

ln P∂

∂gk

ln P

fl

. (11)

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Bound for Local Measurements

Nonlocal measurements (direct detection, shared-entanglement): ||F || ∼ ǫ.

A necessary condition for local (LOCC) measurement is the PPT condition applied tothe POVM [Terhal et al., PRL 86, 5807 (2001)]. Then ||F || ≤ ǫ2 + O(ǫ3).

Generalizable to repeated LOCC measurements

Quantum nonlocality in measurement of nature, even if the state has no entanglement.

M. Tsang, “Quantum nonlocality in weak-thermal-light interferometry,” e-printarXiv:1108.1829.

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Multimode Quantum Sensing: Optical Imaging

M. Tsang, PRA 75, 043813 (2007); PRL 101, 033602(2008).

M. Tsang, PRL 102, 253601 (2009):

Experimentally demonstrated by H. Shin et al.,“Quantum Spatial Superresolution by Optical CentroidMeasurements,” PRL 107, 083603 (2011).

Classical Imaging:

M. Tsang and D. Psaltis, “Reflectionless evanescent wave amplification via twodielectric planar waveguides,” Optics Letters 31, 2741 (2006)

M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” OpticsExpress 15, 11959 (2007)

M. Tsang, “Magnifying perfect lens and superlens design by coordinatetransformation,” Physical Review B 77, 035122 (2008)

L. Waller, M. Tsang, et al., “Phase and amplitude imaging from noisy images byKalman filtering,” Optics Express 19, 2805 (2011).

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Conclusion

Quantum Limit : Dynamical QCRB

Estimation : Quantum Smoothing

Quantum Control : QNC

Applications : phase estimation, force sensing,magnetometry, imaging, etc.

CollaboratorsTheory: Carl Caves, Howard Wiseman

Quantum optics experiment: Elanor Huntington andTrevor Wheatley (UNSW@ADFA), Leonid Krivitskiy (DSI)

Cavity electro-optics: Aaron Danner (LiNbO3), HyunsooYang (superconducting microwave resonator) (ECE, NUS)

Coming postdocs: Ranjith Nair (Horace Yuen’s student),Brent Yen (Jeff Shapiro), Andy Chia (Howard Wiseman)

eletmk@nus.edu.sg

http://mankei.tsang.googlepages.com/

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