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Quantum Theory of Superresolution for Incoherent Optical Imaging ∗

Ranjith Nair, Xiao-Ming Lu, Shan Zheng Ang, and Mankei Tsang

mankei@nus.edu.sg

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

Feb 2017

∗This work is supported by the Singapore National Research Foundation under NRF Award No. NRF-NRFF2011-07 and an MOETier 1 grant.

Quantum Metrology for Dynamical Systems

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■ Generalized “Energetic” Quantum Limits

◆ Decision-theoretic, valid for any POVM◆ Waveform estimation: Tsang, Wiseman, Caves, PRL 106,

090401 (2011).◆ Hypothesis testing : Tsang, Nair, PRA 86, 042115 (2012).◆ Lossy : Tsang, NJP 15, 073005 (2013).◆ Experiment: Iwasawa et al., PRL 111, 163602 (2013).◆ “Heisenberg” limit: Berry, Tsang, Hall, Wiseman, PRX 5,

031018 (2015).◆ Spectrum parameter (e.g., stochastic background), The-

ory+Experiment: Ng et al., PRA 93, 042121 (2016).

■ Quantum-Mechanics-Free Subsystems

◆ a.k.a. Backaction Evasion, Quantum Noise Cancellation, beatbackaction-enforced SQL

◆ Tsang, Caves, PRL 105, 123601 (2010); PRX 2, 031016(2012).

■ Time-Series Analysis

◆ Quantum Smoothing: Tsang, PRL 102, 250403 (2009); PRA80, 033840 (2009); 81, 013824 (2010).

◆ Hypothesis testing : Tsang, PRL 108, 170502 (2012).◆ Fast filter : Tsang, PRA 92, 062119 (2015).

■ Cavity Quantum Electro-Optics

◆ Tsang, PRA 81, 063837 (2010); 84, 043845 (2011).

Quantum Metrology for Incoherent Imaging

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(a)

(b)

Quantum Measurement

on image plane

Tsang, Nair, and Lu, Physical Review X 6, 031033 (2016).Follow-up: Nair and Tsang, Opt. Express 24, 3684 (2016); Tsang, Nair, and Lu, Proc. SPIE 10029, 1002903 (2016); Nair and Tsang,

PRL (Editors’ Suggestion) 117, 190801 (2016); Tsang, arXiv:1605.03799 (2016); Ang, Nair, and Tsang, arXiv:1606.00603 (2016);

Tsang, arXiv:1608.03211 (2016); Lu, Nair, Tsang, arXiv:1609.03025 (2016).

Experiments

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■ Tang, Durak, and Ling, “Fault-tolerant andfinite-error localization for point emitterswithin the diffraction limit,” Optics Express24, 22004 (2016).

■ Yang, Taschilina, Moiseev, Simon, Lvovsky,“Far-field linear optical superresolution viaheterodyne detection in a higher-order localoscillator mode,” Optica 3, 1148 (2016).

■ Tham, Ferretti, Steinberg, “BeatingRayleigh’s Curse by Imaging Using PhaseInformation,” Phys. Rev. Lett., e-printarXiv:1606.02666 (2016).

■ Paúr, Stoklasa, Hradil, Sánchez-Soto, Re-hacek, “Achieving the ultimate optical reso-lution,” Optica 3, 1144 (2016).

Imaging of One Point Source

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Diffraction-Limited

1. Fluorescence Microscopy ($X billion/year)2. Space telescopes (Webb, $8.8 billion)3. Ground-based telescopes:

(a) Large Binocular Telescope (LBT) ($120million)

(b) Giant Magellan Telescope (GMT) (∼$1billion)

(c) Thirty Meter Telescope (TMT) (∼$1.2billion)

(d) European Extremely Large Telescope(E-ELT) (∼$1.2 billion)

LBT Point-Spread Function

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■ Esposito et al., “Large Binocular Telescope Adaptive Optics System: New achievements andperspectives in adaptive optics,” Proc. SPIE 8149, 814902 (2011)

■ Strehl ratio > 80% at infrared

Inferring Position of One Point Source

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■ Classical source■ Given N detected photons, mean-square error:

∆X21 =σ2

N, σ ∼ λ

N.A.. (1)

■ Farrell (1966), Helstrom (1970), Lindegren(1978), Bobroff (1986), ...

■ Full EM, full quantum: Tsang, Optica 2, 646(2015).

Superresolution Microscopy

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■ PALM, STORM, etc.: isolate emitters. Locate cen-troids.

https://cam.facilities.northwestern.edu/588-2/single-molecule-localization-microscopy/

■ Special fluorophores■ slow■ doesn’t work for stars

Two Point Sources

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(a)

(b)

■ Λ(x) = 12

[

|ψ1(x)|2 + |ψ2(x)|2]

■ Rayleigh’s criterion (1879): requires θ2 &σ (heuristic)

Centroid and Separation Estimation

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■ Bettens et al., Ultramicroscopy 77, 37 (1999);Van Aert et al., J. Struct. Biol. 138, 21 (2002);Ram, Ward, Ober, PNAS 103, 4457 (2006).

■ Incoherent sources, Poisson statistics■ X1 = θ1 − θ2/2, X2 = θ1 + θ2/2.■ Cramér-Rao bound for centroid:

∆θ21 ≥σ2

N. (2)

■ CRB for separation estimation: two regimes

◆ θ2 ≫ σ:

∆θ22 ≥4σ2

N, (3)

◆ θ2 ≪ σ:

∆θ22 →4σ2

N×∞ (4)

(a)

(b)

Rayleigh’s Curse

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■ Cramér-Rao bound (standard in single-molecule imaging):

∆θ21 ≥1

J (direct)11∆θ22 ≥

1

J (direct)22(5)

J (direct) is Fisher information for direct imaging■ Gaussian PSF, similar behavior for other PSF■ Rayleigh’s curse■ PALM/STED/STORM: avoid violating Rayleigh

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ

2)

0

0.5

1

1.5

2

2.5

3

3.5

4Classical Fisher information

J(direct)11

J(direct)22

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bound on separation error

Direct imaging (1/J(direct)22 )

Quantum Metrology

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■ Measuring the spatial intensity (direct imaging, CCD) isjust one measurement method. Quantum mechanics al-lows infinite possibilities.

■ Helstrom (1967): For any measurement

Σ ≥ J−1 ≥ K−1, (6)

Kµν =M Re (trLµLνρ) , (7)∂ρ

∂θµ=

1

2(Lµρ+ ρLµ) . (8)

■ K(ρ) is the quantum Fisher information, the ultimateamount of information in the photons.

■ Coherent sources: Tsang, Optica 2, 646 (2015).■ Mixed states:

ρ =∑

n

Dn |en〉 〈en| , (9)

Lµ = 2∑

n,m;Dn+Dm 6=0

〈en| ∂ρ∂θµ |em〉Dn +Dm

|en〉 〈em| . (10)Quantum Measurement

on image plane

Quantum Optics

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■ Mandel and Wolf, Optical Coherence and Quantum Optics; Goodman, Statistical Optics■ Thermal sources, e.g., stars, fluorescent particles.■ Average photon number per mode ǫ≪ 1 at optical frequencies (visible, UV, X-ray, etc.).■ ǫ ∼ 0.01 for the sun at visible, ǫ ∼ 10−6 for fluorophores.

image plane

■ Quantum state in M temporal modes on image plane is ρ⊗M , where

ρ = (1− ǫ) |vac〉 〈vac|+ ǫ2(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|) +O(ǫ2) 〈ψ1|ψ2〉 6= 0, (11)

|ψ1〉 =∫ ∞

−∞

dxψ(x−X1) |x〉 , |ψ2〉 =∫ ∞

−∞

dxψ(x−X2) |x〉 . (12)

■ derive from zero-mean Gaussian P function, mutual coherence■ Multiphoton coincidence: rare, little info as ǫ≪ 1 (homeopathy)■ Similar model for stellar interferometry in Gottesman, Jennewein, Croke, PRL 109, 070503

(2012); Tsang, PRL 107, 270402 (2011).

Plenty of Room at the Bottom

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θ2/σ0 2 4 6 8 10

Fisher

inform

ation

/(N

/4σ2)

0

1

2

3

4Quantum and classical Fisher information

K11

J(direct)11

K22

J(direct)22

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

■ Tsang, Nair, and Lu, Physical Review X 6, 031033 (2016)

∆θ22 ≥1

K22=

1

N∆k2. (13)

■ thermal sources with arbitrary ǫ: Nair and Tsang, PRL (Editors’ Suggestion) 117, 190801(2016); Lupo and Pirandola, ibid. 117, 190802 (2016).

■ Hayashi ed., Asymptotic Theory of Quantum Statistical Inference; Fujiwara JPA 39, 12489(2006): there exists a POVM such that ∆θ2µ → 1/Kµµ, N → ∞.

Hermite-Gaussian Basis

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■ project in Hermite-Gaussian basis:

E1(q) = |φq〉 〈φq | , (14)

|φq〉 =∫ ∞

−∞

dxφq(x) |x〉 , (15)

φq(x) =

(

1

2πσ2

)1/4

Hq

(

x√2σ

)

exp

(

− x2

4σ2

)

. (16)

■ Assume PSF ψ(x) is Gaussian (common).

1

J (HG)22=

1

K22=

4σ2

N. (17)

■ Maximum-likelihood estimator can saturate the classical bound asymptotically for large N .■ arXiv:1605.03799v2: Using SPADE and Max-Like, it is proven that

Σ ≤ 16σ2

Nfor any detected N. (18)

Spatial-Mode Demultiplexing (SPADE)

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image plane

...

...

image plane

...

...

■ Many other ways (optical comm.), e.g.,

◆ DAB Miller, “Self-configuring universal linear optical components,” Photonics Research 1, 1(2013).

◆ Li et al., “Space-division multiplexing: the next frontier in optical communication,”Adv. Opt. Photon. 6, 413 (2014).

Elementary Explanation

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■ Incoherent sources: energy in first-order mode is ∝ (d/2)2 + (−d/2)2 = d2/2■ Zeroth-order mode is just background noise, removing it improves SNR.■ Why quantum formalism?

◆ Fundamental quantum limit◆ Ensures measurement is physical◆ Discover new possibilities

SLIVER (SuperLocalization via Image-inVERsion interferometry)

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Estimator

Image

Inversion

■ Nair and Tsang, Opt. Express 24, 3684 (2016).■ Laser Focus World, Feb 2016 issue.■ Prior classical theory/experiment of image-inversion interferometer:

◆ Wicker, Heintzmann, Opt. Express 15, 12206 (2007).◆ Wicker, Sindbert, Heintzmann, Opt. Express 17, 15491 (2009).◆ Weigel, Babovsky, Kiessling, Kowarschik, Opt. Commun. 284, 2273 (2011).◆ No statistical analysis, predicted

Theoretical Follow-up

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Dimensions Sources Theory Experimental proposals1. Tsang, Nair, and Lu,

Phys. Rev. X 6, 031033(2016)

1D Weak thermal (opticalfrequencies and above)

Quantum SPADE

2. Nair and Tsang, Op-tics Express 24, 3684(2016)

2D Thermal (any fre-quency)

Semiclassical SLIVER

3. Tsang, Nair, and Lu,Proc. SPIE 10029,1002903 (2016)

N/A Weak thermal, lasers Semiclassical N/A

4. Nair and Tsang, PRL(Editors’ Suggestion)117, 190801 (2016)

1D Thermal Quantum SLIVER

5. Tsang,arXiv:1605.03799

1D Weak thermal Quantum, Bayesian,Minimax

SPADE

6. Ang, Nair, and Tsang,arXiv:1606.00603

2D Weak thermal Quantum SPADE, SLIVER

7. Tsang,arXiv:1608.03211

2D Weak thermal, multiplesources

Quantum SPADE

8. Lu, Nair, Tsang,arXiv:1609.03025

2D Weak thermal, one-versus-two

Quantum, binary hy-pothesis testing

SPADE, SLIVER

Other groups:

■ Lupo and Pirandola, PRL (Editors’ Suggestion) 117, 190801 (2016).■ Rehacek et al., Optics Letters 42, 231 (2017).■ Krovi, Guha, Shapiro, arXiv:1609.00684.

Experiments

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■ Tang, Durak, and Ling, “Fault-tolerant and finite-error localization for point emitters within thediffraction limit,” Optics Express 24, 22004 (2016).

◆ SLIVER◆ Laser, classical noise

■ Yang, Taschilina, Moiseev, Simon, Lvovsky, “Far-field linear optical superresolution viaheterodyne detection in a higher-order local oscillator mode,” Optica 3, 1148 (2016).

◆ Mode heterodyne◆ Laser

■ Tham, Ferretti, Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,”Phys. Rev. Lett., e-print arXiv:1606.02666 (2016).

◆ variation of SPADE◆ two independent SPDC sources, ∼ 2× quantum limit

■ Paúr, Stoklasa, Hradil, Sánchez-Soto, Rehacek, “Achieving the ultimate optical resolution,”Optica 3, 1144 (2016).

◆ variation of SPADE◆ laser, close to quantum limit (caveat)

SPLICE

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■ Tham, Ferretti, Steinberg, Phys. Rev. Lett., e-print arXiv:1606.02666 (2016).

Popular Coverage

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■ Viewpoint in APS Physics■ IoP Physics World and nanotechweb.org:

Seth Lloyd of the Massachusetts Instituteof Technology in the US is impressed. ‘This isawesome work and I am amazed that it hasn’tbeen done before,’ he says. ‘Perhaps everyonethought it was too good to be true.’

■ APS Physics Central■ Phys.org

Arbitrary Source Distributions

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3

iTEM

21

3

2

1

×104

0

2

1.5

1

0.5

|θ10|10−3 10−2 10−1

MSE

forθ̌10

×10−5

2

4

6

8X moment

|θ01|10−3 10−2 10−1

MSE

forθ̌01

×10−5

2

4

6

8Y moment

θ20

10−2 10−1

MSE

forθ̌20

10−6

10−5

10−4X2 moment

θ02

10−2 10−1

MSE

forθ̌02

10−6

10−5

10−4Y 2 moment

β(10, 01) ×10−35 10 15

MSE

forθ̌11

10−6

10−5

XY moment

SPADE (simulated)

SPADE (theory)

direct imaging (simulated)

direct imaging (theory)

■ Yang et al., Optica 3, 1148 (2016): even moments■ Tsang, arXiv:1608.03211: Generalized SPADE: Enhanced estimation of 2nd or higher moments

Quantum Metrology Kills Rayleigh’s Criterion

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θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramér-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

image plane

...

...

Estimator

Image

Inversion

■ FAQ: https://sites.google.com/site/mankeitsang/news/rayleigh/faq■ email: mankei@nus.edu.sg

https://sites.google.com/site/mankeitsang/news/rayleigh/faqmankei@nus.edu.sg

ǫ ≪ 1 Approximation

25 / 32

■ Chap. 9, Goodman, Statistical Optics:

“If the count degeneracy parameter is much less than 1, it is highly probable that there will be either zero or

one counts in each separate coherence interval of the incident classical wave. In such a case the classical

intensity fluctuations have a negligible ”bunching” effect on the photo-events, for (with high probability) the

light is simply too weak to generate multiple events in a single coherence cell.

■ Zmuidzinas (https://pma.caltech.edu/content/jonas-zmuidzinas), JOSA A 20, 218 (2003):

“It is well established that the photon counts registered by the detectors in an optical instrument follow

statistically independent Poisson distributions, so that the fluctuations of the counts in different detectors are

uncorrelated. To be more precise, this situation holds for the case of thermal emission (from the source, the

atmosphere, the telescope, etc.) in which the mean photon occupation numbers of the modes incident on the

detectors are low, n ≪ 1. In the high occupancy limit, n ≫ 1, photon bunching becomes important in that it

changes the counting statistics and can introduce correlations among the detectors. We will discuss only the

first case, n ≪ 1, which applies to most astronomical observations at optical and infrared wavelengths.”

■ Hanbury Brown-Twiss (post-selects on two-photon coincidence, homeopathy): poor SNR,obsolete for decades in astronomy.

■ See also Labeyrie et al., An Introduction to Optical Stellar Interferometry, etc.■ Fluorescent particles: Pawley ed., Handbook of Biological Confocal Microscopy, Ram, Ober,

Ward (2006), etc., may have antibunching, but Poisson model is fine and standard because ofǫ≪ 1.

Binary SPADE

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image plane

image plane

leaky modes

leaky modes

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ

2)

0

0.2

0.4

0.6

0.8

1Classical Fisher information

J(HG)22 = K22

J(direct)22

J(b)22

θ2/W0 1 2 3 4 5

Fisher

inform

ation/(π

2N/3W

2)

0

0.2

0.4

0.6

0.8

1Fisher information for sinc PSF

K22

J(direct)22

J(b)22

Numerical Performance of Maximum-Likelihood Estimators

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θ2/σ0 0.5 1 1.5 2

Mean-squareerror/(4σ2/L

)

0

0.5

1

1.5

2Simulated errors for SPADE

1/J′(HG)22 = 1/K

′22

L = 10L = 20L = 100

θ2/σ0 0.5 1 1.5 2

Mean-squareerror/(4σ2/L

)

0

0.5

1

1.5

2Simulated errors for binary SPADE

1/J′(b)22

L = 10L = 20L = 100

■ L = number of detected photons■ biased (violate CRB), < 2×CRB.

Minimax/Bayesian

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Bayesian CRB for any biased/unbiased estimator (e-print arXiv:1605.03799v2)

Quantum/SPADE: supθ

Σ22(θ) ≥4σ2

N, Direct imaging: sup

θΣ

(direct)22 (θ) ≥

√2σ2

3√N. (19)

Photon Number L103 104

supθMSE

/σ2

10−4

10−3

10−2

10−1(d) Worst-case error

Quantum/SPADE limit

SPADE (ML)

SPADE (modified ML)

Direct-imaging limit

Direct imaging (ML)

θ/σ0 0.1 0.2 0.3

MSE

/(4σ2/L)

0

0.5

1

1.5

2(a) SPADE with ML

CRBL=1000L=2000L=5000L=10000

θ/σ0 0.1 0.2 0.3

MSE

/(4σ2/L)

0

0.5

1

1.5

2(b) SPADE with modified ML

CRBL=1000L=2000L=5000L=10000

θ/σ0 0.1 0.2 0.3

MSE

/(4σ2/L)

0

20

40

60(c) Direct imaging with ML

CRBL=1000L=2000L=5000L=10000

2D SPADE and SLIVER

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Ang, Nair, Tsang, e-print arXiv:1606.00603

Misalignment

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SPADE

photon-counting

array

image

plane

image

plane

object

plane

■ ξ ≡ |θ̌1 − θ|/σ ≪ 1■ Overhead photons N1 ∼ 1/ξ2■ ξ = 0.1, N1 ∼ 100.■ CRB for Xs = θ1 ± θ2/2

θ2/σ0 2 4 6 8 10

Fisher

inform

ation/(N

/4σ2)

0

0.2

0.4

0.6

0.8

1Fisher information for misaligned SPADE

J(HG)22 (ξ = 0)

J(HG)22 (ξ = 0.1)

J(HG)22 (ξ = 0.2)

J(HG)22 (ξ = 0.3)

J(HG)22 (ξ = 0.4)

J(HG)22 (ξ = 0.5)

J(direct)22

θ2/σ0 0.5 1 1.5 2

mean-square

error/(4σ2/L)

0

2

4

6

8

10

12Simulated errors for misaligned binary SPADE

1/J′(b)22 (ξ = 0.1)

1/J′(direct)22

L = 10L = 100L = 1000

10−1 100 101

Mean-square

error/(2σ2/N

)

100

101

102Cramér-Rao bounds on localization error

Hybrid (ξ = 0)Hybrid (ξ = 0.1)Hybrid (ξ = 0.2)Hybrid (ξ = 0.3)Hybrid (ξ = 0.4)Hybrid (ξ = 0.5)Direct imaging

Quantum Computational Imaging

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■ Design quantum computer to

◆ Maximize information extraction◆ Reduce classical computational complexity

Quantum Technology 1.5

32 / 32

Quantum Metrology for Dynamical SystemsQuantum Metrology for Incoherent ImagingExperimentsImaging of One Point SourceLBT Point-Spread FunctionInferring Position of One Point SourceSuperresolution MicroscopyTwo Point SourcesCentroid and Separation EstimationRayleigh's CurseQuantum MetrologyQuantum OpticsPlenty of Room at the BottomHermite-Gaussian BasisSpatial-Mode Demultiplexing (SPADE)Elementary ExplanationSLIVER (SuperLocalization via Image-inVERsion interferometry)Theoretical Follow-upExperimentsSPLICEPopular CoverageArbitrary Source DistributionsQuantum Metrology Kills Rayleigh's Criterion1 ApproximationBinary SPADENumerical Performance of Maximum-Likelihood EstimatorsMinimax/Bayesian2D SPADE and SLIVERMisalignmentQuantum Computational ImagingQuantum Technology 1.5