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Resolving Starlight: A Quantum Perspective

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

National University of Singapore

IPS, 1 Oct 2021

Fundamental Resolution of Incoherent Imaging

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Observational Astronomy Fluorescence Microscopy(images from the internet)

Quantum information theory Quantum-inspired measurements

Wave Nature of Light: Diffraction Limit

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Diffraction-limited (λ/NA)

1. Fluorescence microscopy2. Space telescopes (Webb, $10 billion)3. Ground-based telescopes (corrected by adap-

tive optics):

(a) Large Binocular Telescope (LBT) (Strehlratio > 80%, $120 million)

(b) Giant Magellan Telescope (GMT)(c) Thirty Meter Telescope (TMT)(d) European Extremely Large Telescope (E-

ELT) (>$1 billion each)

Esposito et al., SPIE 8149, 814902 (2011).

Rayleigh (1879)

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Resolved:

Not resolved:

Rough Idea

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“Rayleigh’s criterion is a rough idea...” a better resolutioncan be achieved “if sufficiently careful measurements ofthe exact intensity distribution over the diffracted imagespot can be made.”

Chapter 30. Diffraction, Feynman Lectures on Physics, Vol. I

Photon Shot Noise

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Poisson at optical frequencies Bunching (thermal)/anti-bunching (fluor.) negligible in practice Random arrival of “photons”

Parameter Estimation

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Estimator θ(Y ): guess θ from noisy data Y

Mean-square error: MSE = E[θ(Y )− θ

]2.

Cramer-Rao bound (unbiased estimators):

MSE(θ) ≥ J(θ)−1, J(θ) = N

∫ ∞

−∞

dxf(x|θ)

[∂

∂θln f(x|θ)

]2

.

J : Fisher information MSE → J(θ)−1 via maximum-likelihood.

Quantum Limit to Passive Imaging

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Quantum Measurement

on image plane

Quantum: Direct imaging is just one of the in-finitely many possible measurements.

Born’s rule: P (y|θ) = trE(y)ρ(θ) Helstrom (1967), Nagaoka (1989), etc.: For any

POVM,

MSE ≥ J−1 ≥ K−1,

K(ρ⊗M ) =M trS2ρ,

∂ρ

∂θ=

1

2(Sρ+ ρS) .

K(ρ) is the quantum Fisher information, the ulti-mate amount of information in the photons.

Plenty of Room at the Bottom

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θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramer-Rao bounds on separation error

Quantum (1/K22)

Direct imaging (1/J(direct)22 )

Tsang, Nair, Lu, PRX 6, 031033 (2016).

Optimal Measurement

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Sort the photons in Hermite-Gaussian(TEM) modes first, then do photoncounting

Spatial-mode demultiplexing (SPADE)

– Tsang, Nair, Lu, PRX (2016); Rehaceket al., OL 42, 231 (2017).

Many implementations (in optical comm.,photonic circuits, interferometers, spatiallight modulators, etc.)

Classical sources Linear optics/photon counting Killer applications (astronomy, fluores-

cence microscopy, etc.)

Elementary Explanation

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Incoherent sources: energy in 1st-order mode is

∝

(d

2

)2

+

(

−d

2

)2

=d2

2.

0th-order mode is just background noise; filtering it improves SNR. Incoherence (sources) + Coherence (diffraction) + Signal-dependent noise

(Poisson)

Experimental Demonstrations

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1. Tham, Ferretti, Steinberg (Toronto),PRL 118, 070801 (2017).

MSE ∼ 5× QCRB ∼ 1

9 × MSE of direct imaging(biased estimator)

2. Tang, Durak, Ling (CQT Singapore), OE 24, 22004 (2016).3. Yang, Taschilina, Moiseev, Simon, Lvovsky (Calgary),

Optica 3, 1148 (2016).4. Paur, Stoklasa, Hradil, Sanchez-Soto, Rehacek

(Palacky/Madrid/Max Planck), Optica 3, 1144 (2016).

5. Parniak et al. (Warsaw), PRL 121, 250503 (2018).6. Donohue et al. (Paderborn), PRL 121, 090501 (2018).7. Paur et al. (Palacky/Madrid/Max Planck/ESA), Optica 5,

1177 (2018).8. J. Hassett et al. (Rochester), FiO/LS, JW4A.124 (2018).9. Zhou et al. (Rochester), Optica 6, 534 (2019).

10. Paur et al., OL 44, 3114 (2019).11. Wadood et al., (Rochester) Fio/LS, FM3C.7 (2019).12. Rehacek et al., PRL 123, 193601 (2019).13. Salit et al. (Honeywell), AO 59, 5319 (2020).14. Zhang et al. (Stevens), OL 45, 4968 (2020).15. Boucher et al. (Kastler Brossel), Optica 7, 1621 (2020).16. Ansari et al. (Paderborn), PRXQ 2, 010301 (2020).17. Brecht et al. (Paderborn), OSA Quantum 2.0, QW6A.17

(2020).18. Wadood et al., Optics Express 29, 22034 (2021).19. Mouradian et al. (Berkeley), PRA 103, 032419 (2021).20. De et al. (Paderborn), PRR 3, 033082 (2021).21. Santra et al. (Iowa State), JPCB 125, 3092 (2021).22. Pushkina et al. (Oxford), arXiv:2105.01743 (2021).

23. Mazelanik et al. (Warsaw), arXiv:2106.04450 (2021).

Other Implementations

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Tang et al. (CQT), OE (2016), etc.

Boucher et al. (Cailabs), Optica(2020) https://www.youtube.com/watch?v=fKmXakWRUjU

Ansari et al. (Paderborn), PRXQ (2020)

Salit et al. (Honeywell), AO (2020).

SPADE for multiple sources

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Wavefunction from each point source:

For one point source,

Energy in first-order mode ∝ X2.

A distribution of many incoherent sources:

Total energy in first-order mode ∝

∫

dXF (X)X2.

Higher-order modes measure higher-order moments.

Our Publications

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1. Tsang, Nair, Lu, PRX 6, 031033 (2016).2. Nair, Tsang, OE 24, 3684 (2016).3. Tsang, Nair, Lu, SPIE 10029, 1002903 (2016).4. Nair, Tsang, PRL 117, 190801 (2016).5. Ang, Nair, Tsang, PRA 95, 063847 (2017).6. Tsang, NJP 19, 023054 (2017).7. Yang, Nair, Tsang, Simon, Lvovsky, PRA 96, 063829 (2017).8. Tsang, JMO 65, 104 (2018).9. Tsang, PRA 97, 023830 (2018).10. Lu, Krovi, Nair, Guha, Shapiro, npj Quantum Information 4, 64 (2018).11. Tsang, PRA 99, 012305 (2019).12. Tsang, Nair, Optica 6, 400 (2019).13. Tsang, PRR 1, 033006 (2019).14. Tsang, arXiv:2010.11084 (2020).15. Tsang, arXiv:2010.03518 (2021).16. Tsang, Quantum 5, 527 (2021).17. Review paper: Tsang, Contemporary Physics 60, 279 (2019).

Other references: https://blog.nus.edu.sg/mankei/superresolution/

Conclusion

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Quantum information theory reveals hiddeninformation in starlight, fluorescence

Spatial-mode demultiplexing (SPADE) im-proves subdiffraction imaging.

National Research Foundation Singapore

– Quantum Engineering Program (QEP-P7):experimental fluorescence microscopy

mankei@nus.edu.sg,https://blog.nus.edu.sg/mankei/

Quantum Technology 1.5

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Misalignment

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∆ = centroid displacement+ object size ≪ 1

Adaptive measurements: Grace et al. (Arizona), JOSAA 37, 1288 (2020).

Stellar Interferometry

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Interferometers

far field

Same principles apply to stellar interferometry if we back-propagate. Change the OTF (ψ(x) → Ψ(k)) to match the aperture function.

Stellar Interferometry

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Astrophotonics: photonic circuitsfor stellar interferometry

“Dragonfly,” Jovanovic et al.,Mon. Not. R. Astron. Soc. 427, 806(2012)

Conventional wisdom: robust againstatmospheric turbulence, can’t com-pete with direct imaging at diffrac-tion limit; see, e.g.,

– Goodman, Statistical Optics– Zmuidzinas, JOSA A (2003):

It is important to remember that

the imperfect beam patterns of

sparse-aperture interferometers

extract a sensitivity penalty as

compared with filled-aperture tele-

scopes, even after accounting for

the differences in collecting areas.

Our work:

– Fundamental advantage withdiffraction + photon shot noise

– Proper statistics– Quantum limit

Photon Shot Noise

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Thermal sources (stars, etc.)

– Poisson, bunching negligible at optical– Goodman, Statistical Optics; Zmuidzinas, JOSA A

20, 218 (2003)

Fluorophores (GFP, dye molecules, quantum dots,etc.)

– Poisson, negligible anti-bunching– Pawley ed., Handbook of Biological Confocal Mi-

croscopy ; Ram, Ober, Ward, PNAS (2006)

Estimating Two-Point Separation

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Conventional direct imaging (photon counting on image plane, Poisson noise):

J(0) = 0, J(∞) = N4σ2 , σ = λ

NA

θ2/σ0 0.2 0.4 0.6 0.8 1

Mean-squareerror/(4σ2/N

)

0

20

40

60

80

100Cramer-Rao bound on separation error

Direct imaging (1/J(direct)22 )

1J(0) = ∞, 1

J(∞) =4σ2

N

“Rayleigh’s curse” See, e.g., Ram, Ward, Ober, PNAS 103, 4457 (2006).

Superresolution Microscopy

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PALM, STORM, STED, etc.: make sparsesubsets of fluorophores emit

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

avoid violating Rayleigh Need controllable fluorophores slow, phototoxicity doesn’t work for stars, passive imaging

Quantum Optics for Incoherent Imaging

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Thermal optical source: average photon number per mode ǫ≪ 1

image plane

Quantum state in M spectral modes = ρ⊗M , N =Mǫ

ρ = (1− ǫ) |vac〉 〈vac|+ǫ

2(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|) +O(ǫ2),

|ψs〉 ≡

∫ ∞

−∞

dxψ(x−Xs) |x〉 .

derived from Glauber representation. Consistent with Poisson counting statistics. see, e.g., Tsang, PRL 107, 270402 (2011); Tsang, Nair, Lu, PRX (2016);

Quantum (2021).

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Part II: Extended Sources

Arbitrary Source Distribution

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Direct imaging of UNKNOWN/INFINITE number of incoherent pointsources

Object intensity = F (X|θ).

f(x|θ) =

∫

dX|ψ(x−X)|2F (X|θ), Jµν = N

∫ ∞

−∞

dxf

(∂

∂θµln f

)(∂

∂θνln f

)

.

MSEµµ ≥ CRBµ ≡ (J−1)µµ.

F unknown: Infinitely many scalar parameters (infinite-dimensional)

Semiparametric Estimation

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Bickel et al., Efficient and Adaptive Estimation for SemiparametricModels:

Each scalar θj → score function Sj ≡∂

∂θjln f(x|θ).

CRB in terms of geometric operations:

– Sj ∈ L2(f) (Hilbert space, 〈a, b〉 ≡∫a(x)b(x)f(x|θ)dx).

– tangent space: T ≡ spanSj ⊂ L2(f).– Define scalar parameter of interest β(θ), error (“influence function”)

δ ≡ β(x)− β(θ) ∈ L2(f).– CRB = ‖Π(δ|T )‖2 (Projection into tangent space).

Defining the Subdiffraction Regime

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Sparse (Good Regime,PALM, STED, compressedsensing, etc.)

Subdiffraction (Worst-CaseRegime)

object width (relative to origin) ≡ ∆ ≪ 1.

Cramer-Rao Bound for Direct Imaging

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Define the parameter of interest as an object moment:

β =

∫

dXF (X)Xµ, µ = integer

Assume F (X) unknown (semiparametric) ∆ ≪ 1:

CRB(direct) =Ω(∆)

N.

Signal-to-Noise Ratio:

β = O(∆µ),β2

CRB(direct)= NO(∆2µ−1). (1)

Tsang, PRR 1, 033006 (2019); arXiv:2010.03518 (2021).

Quantum Semiparametric Estimation

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Each scalar θj → symmetric logarithmic derivative ∂ρ∂θj

= Sj ρ ≡ 12 (Sjρ+ ρSj).

QCRB in terms of geometric operations:

– Sj ∈ L2(ρ) (Hilbert space, 〈a, b〉 ≡ tr(a b)ρ (Holevo)).– tangent space: T ≡ spanSj ⊂ L2(ρ).– Define parameter of interest β(θ), error (“influence operator”)

δ ≡∫β(x)E(dx)− β(θ) ∈ L2(ρ) (E = POVM).

– QCRB = ‖Π(δ|T )‖2.

Tsang, Albarelli, Datta, PRX (2020).

Quantum Limit

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One-photon density operator:

ρ1(F ) =

∫

dXF (X)e−ikX |ψ〉 〈ψ| eikX , |ψ〉 =

∫ ∞

−∞

dxψ(x) |x〉 .

– mixed state– F unknown (semiparametric)– infinite rank (infinite number of spatial modes)

Quantum bound [rigorous: Tsang, arXiv:2010.03518 (2021)]:

β2

QCRB= NO(∆2⌈µ/2⌉), compare with

β2

CRB(direct)= NO(∆2µ−1).

See also Tsang, PRA (2019); Zhou & Jiang (Yale), PRA (2019). Big enhancements over direct imaging possible when

– Subdiffraction: ∆ ≪ 1– Second or higher moments: µ ≥ 2

SPADE can get close (order of magnitude).

Generalized SPADE for Moment Estimation in 2D Imaging

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Gaussian PSF [Tsang, NJP (2017)]:

– For moments with even µ1 & even µ2: TEM basis

⊲ See also Yang et al., Optica (2016)

– For other moments: interference of pairs of TEM modes

More General PSFs

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Any centrosymmetric and separable PSF [Tsang, PRA (2018)]:

– For moments with even µ1, even µ2: “PSF-adapted” (PAD) basis(Rehacek et al. OL (2017), generalizes TEM)

– For other moments: interference of pairs of PAD modes

PAD

iPAD1 iPAD2 iPAD3

iPAD4 iPAD5 iPAD6

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01

02

03

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13 23 33 00 10 20 30

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03

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03

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02

03

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02

03

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03

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(10) (50)

(12) (52)

(54)

(56)

(14)

(16)

(01) (21) (41) (61)

(05) (25) (45) (65)

(11) (31) (51)

(15) (35) (55)

(36)

(34)

(32)

(30)

(03) (23) (43) (63) (13) (53)(33)

(00) (20) (40) (60)

(02) (22) (42) (62)

(04) (24) (44) (64)

(06) (26) (46) (66)

Performance of Generalized SPADE

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MSE of SPADE:

MSE =Θ(∆2⌊µ/2⌋

)

N.

∼ quantum limit, big enhancement overdirect imaging when

– ∆ ≪ 1 (subdiffraction)– µ = µX + µY ≥ 2

Caveat: β2 = O(∆2µ), SNR:

β2

MSE= NO(∆2⌈µ/2⌉).

Need many photons, especially for large µ.

|θ1| / θ0

10−2 10−1

MSE

/[θ

2 0(∆

/2)2

µ]

×10−3

1

2

3

4

567

µ = 1

θ2 / θ0 ×10−32 4 6

MSE

/[θ

2 0(∆

/2)2

µ]

10−3

10−2

10−1

100µ = 2

θ2 / θ0 ×10−32 4 6

MSE

/[θ

2 0(∆

/2)2

µ]

100

102

µ = 3

θ4 / θ0

10−6 10−5

MSE

/[θ

2 0(∆

/2)2

µ]

100

105µ = 4

Direct imagingDirect imaging (CRB)SPADESPADE (theory)

Tsang, NJP (2017); PRA(2018); PRR (2019).

Superoscillation

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αµ = a0β0 + a1β1 + · · ·+ aµβµ =

∫

(a0 + a1X + · · ·+ aµXµ)

︸ ︷︷ ︸

polynomial

F (X)dX (2)

The polynomial can be an orthogonal polynomial (oscillatory).

(Legendre polynomials, from wikipedia) αµ is like a Fourier coefficient. Resembles superoscillation and Slepian theory Tsang, arXiv:2010.11084 (2020).