Abstract

Motivated by the importance of optical microscopes to science and engineering, scientists have pondered for centuries how to improve their resolution and the existence of fundamental resolution limits. In recent years, a new class of microscopes that overcome a long-held belief about the resolution have revolutionized biological imaging. Termed “super-resolution” microscopy, these techniques work by accurately locating optical point sources from far field. To investigate the fundamental localization limits, here I derive quantum lower bounds on the error of locating point sources in free space, taking full account of the quantum, nonparaxial, and vectoral nature of photons. These bounds are valid for any measurement technique, as long as it obeys quantum mechanics, and serve as general no-go theorems for the resolution of microscopes. To arrive at analytic results, I focus mainly on the cases of one and two classical monochromatic sources with an initial vacuum optical state. For one source, a lower bound on the root-mean-square position estimation error is of the order of λ0/N, where λ0 is the free-space wavelength and N is the average number of radiated photons. For two sources, owing to the statistical effect of nuisance parameters, the error bound diverges when their radiated fields overlap significantly. The use of squeezed light to further enhance the accuracy of locating one classical point source and the localization limits for partially coherent sources and single-photon sources are also discussed. The theory presented establishes a rigorous quantum statistical inference framework for the study of super-resolution microscopy and points to the possibility of using quantum techniques for true resolution enhancement.

© 2015 Optical Society of America

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References

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2014 (2)

L. A. Rozema, J. D. Bateman, D. H. Mahler, R. Okamoto, A. Feizpour, A. Hayat, and A. M. Steinberg, “Scalable spatial superresolution using entangled photons,” Phys. Rev. Lett. 112, 223602 (2014).
[Crossref]

D. Gatto Monticone, K. Katamadze, P. Traina, E. Moreva, J. Forneris, I. Ruo-Berchera, P. Olivero, I. P. Degiovanni, G. Brida, and M. Genovese, “Beating the Abbe diffraction limit in confocal microscopy via nonclassical photon statistics,” Phys. Rev. Lett. 113, 143602 (2014).
[Crossref]

2013 (7)

M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7, 229–233 (2013).
[Crossref]

A. De Pasquale, D. Rossini, P. Facchi, and V. Giovannetti, “Quantum parameter estimation affected by unitary disturbance,” Phys. Rev. A 88, 052117 (2013).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref]

O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, “Superresolution microscopy with quantum emitters,” Nano Lett. 13, 5832–5836 (2013).
[Crossref]

J.-M. Cui, F.-W. Sun, X.-D. Chen, Z.-J. Gong, and G.-C. Guo, “Quantum statistical imaging of particles without restriction of the diffraction limit,” Phys. Rev. Lett. 110, 153901 (2013).
[Crossref]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

M. Tsang, “Quantum metrology with open dynamical systems,” New J. Phys. 15, 073005 (2013).
[Crossref]

2012 (10)

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref]

H. J. Hyvärinen, S. Rehman, J. Tervo, J. Turunen, and C. J. R. Sheppard, “Limitations of superoscillation filters in microscopy applications,” Opt. Lett. 37, 903–905 (2012).
[Crossref]

L. T. Hall, G. C. G. Beart, E. A. Thomas, D. A. Simpson, L. P. McGuinness, J. H. Cole, J. H. Manton, R. E. Scholten, F. Jelezko, J. Wrachtrup, S. Petrou, and L. C. L. Hollenberg, “High spatial and temporal resolution wide-field imaging of neuron activity using quantum NV-diamond,” Sci. Rep. 2, 401 (2012).
[Crossref]

W. P. Dempsey, S. E. Fraser, and P. Pantazis, “SHG nanoprobes: advancing harmonic imaging in biology,” BioEssays 34, 351–360 (2012).
[Crossref]

O. Schwartz and D. Oron, “Improved resolution in fluorescence microscopy using quantum correlations,” Phys. Rev. A 85, 033812 (2012).
[Crossref]

C. A. Pérez-Delgado, M. E. Pearce, and P. Kok, “Fundamental limits of classical and quantum imaging,” Phys. Rev. Lett. 109, 123601 (2012).
[Crossref]

P. R. Hemmer and T. Zapata, “The universal scaling laws that determine the achievable resolution in different schemes for super-resolution imaging,” J. Opt. 14, 083002 (2012).
[Crossref]

R. W. Boyd and J. P. Dowling, “Quantum lithography: status of the field,” Quantum Inf. Process. 11, 891–901 (2012).
[Crossref]

M. Tsang and R. Nair, “Fundamental quantum limits to waveform detection,” Phys. Rev. A 86, 042115 (2012).
[Crossref]

M. Tsang, “Ziv–Zakai error bounds for quantum parameter estimation,” Phys. Rev. Lett. 108, 230401 (2012).
[Crossref]

2011 (5)

M. Tsang, H. M. Wiseman, and C. M. Caves, “Fundamental quantum limit to waveform estimation,” Phys. Rev. Lett. 106, 090401 (2011).
[Crossref]

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref]

R. Nair and B. J. Yen, “Optimal quantum states for image sensing in loss,” Phys. Rev. Lett. 107, 193602 (2011).
[Crossref]

S. Pirandola, “Quantum reading of a classical digital memory,” Phys. Rev. Lett. 106, 090504 (2011).
[Crossref]

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

2009 (6)

F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).
[Crossref]

C.-L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Three-dimensional harmonic holographic microcopy using nanoparticles as probes for cell imaging,” Opt. Express 17, 2880–2891 (2009).
[Crossref]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[Crossref]

M. Tsang, “Quantum imaging beyond the diffraction limit by optical centroid measurements,” Phys. Rev. Lett. 102, 253601 (2009).
[Crossref]

R. Heintzmann and M. G. L. Gustafsson, “Subdiffraction resolution in continuous samples,” Nat. Photonics 3, 362–364 (2009).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inform. 7, 125–137 (2009).
[Crossref]

2008 (3)

H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, “Optical imaging beyond the diffraction limit via dark states,” Phys. Rev. A 78, 013803 (2008).
[Crossref]

Y. Pu, M. Centurion, and D. Psaltis, “Harmonic holography: a new holographic principle,” Appl. Opt. 47, A103–A110 (2008).
[Crossref]

N. I. Zheludev, “What diffraction limit?” Nat. Mater. 7, 420–422 (2008).
[Crossref]

2007 (3)

C. J. R. Sheppard, “The optics of microscopy,” J. Opt. A 9, S1 (2007).
[Crossref]

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104, 12596–12602 (2007).

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[Crossref]

2006 (4)

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

A. Fujiwara, “Strong consistency and asymptotic efficiency for adaptive quantum estimation problems,” J. Phys. A 39, 12489–12504 (2006).
[Crossref]

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[Crossref]

S. Ram, E. S. Ward, and R. J. Ober, “Beyond Rayleigh’s criterion: a resolution measure with application to single-molecule microscopy,” Proc. Natl. Acad. Sci. USA 103, 4457–4462 (2006).

2005 (2)

K. Lidke, B. Rieger, T. Jovin, and R. Heintzmann, “Superresolution by localization of quantum dots using blinking statistics,” Opt. Express 13, 7052–7062 (2005).
[Crossref]

X. Michalet, F. F. Pinaud, L. A. Bentolila, J. M. Tsay, S. Doose, J. J. Li, G. Sundaresan, A. M. Wu, S. S. Gambhir, and S. Weiss, “Quantum dots for live cells, in vivo imaging, and diagnostics,” Science 307, 538–544 (2005).
[Crossref]

2004 (1)

R. J. Ober, S. Ram, and E. S. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[Crossref]

2003 (2)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref]

S. Barnett, C. Fabre, and A. Maıtre, “Ultimate quantum limits for resolution of beam displacements,” Eur. Phys. J. D 22, 513–519 (2003).

2002 (1)

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[Crossref]

2001 (1)

P. Stoica and T. L. Marzetta, “Parameter estimation problems with singular information matrices,” IEEE Trans. Signal Process. 49, 87–90 (2001).
[Crossref]

2000 (4)

A. Rotnitzky, D. R. Cox, M. Bottai, and J. Robins, “Likelihood-based inference with singular information matrix,” Bernoulli 6, 243–284 (2000).
[Crossref]

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurement of very small displacements in optical images,” Opt. Lett. 25, 76–78 (2000).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref]

1999 (3)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[Crossref]

R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100  nm axial resolution,” J. Microsc. 195, 10–16 (1999).
[Crossref]

1996 (2)

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135–173 (1996).
[Crossref]

M. G. Paris, “Displacement operator by beam splitter,” Phys. Lett. A 217, 78–80 (1996).
[Crossref]

1994 (1)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref]

1986 (1)

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57, 1152–1157 (1986).
[Crossref]

1978 (1)

R. W. Miller and C. B. Chang, “A modified Cramér–Rao bound and its applications (corresp.),” IEEE Trans. Inform. Theory 24, 398–400 (1978).
[Crossref]

1973 (1)

H. P. Yuen and M. Lax, “Multiple-parameter quantum estimation and measurement of nonselfadjoint observables,” IEEE Trans. Inform. Theory 19, 740–750 (1973).
[Crossref]

1970 (1)

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
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Ann. Phys. (1)

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Appl. Opt. (1)

Bernoulli (1)

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BioEssays (1)

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Biophys. J. (2)

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Eur. Phys. J. D (1)

S. Barnett, C. Fabre, and A. Maıtre, “Ultimate quantum limits for resolution of beam displacements,” Eur. Phys. J. D 22, 513–519 (2003).

IEEE Trans. Inform. Theory (2)

H. P. Yuen and M. Lax, “Multiple-parameter quantum estimation and measurement of nonselfadjoint observables,” IEEE Trans. Inform. Theory 19, 740–750 (1973).
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IEEE Trans. Signal Process. (1)

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J. Opt. Soc. Am. (1)

J. Phys. A (2)

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Nano Lett. (2)

O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, “Superresolution microscopy with quantum emitters,” Nano Lett. 13, 5832–5836 (2013).
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F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009).
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Figures (3)

Fig. 1.
Fig. 1.

Classical point source with dipole moment p(t) radiating in free space. Its position r is estimated by measuring the quantum optical field, with a(k,s) denoting its annihilation operator.

Fig. 2.
Fig. 2.

Two classical point sources with dipole moments p(t) and p(t) at r and r, respectively, with quantum optical radiation.

Fig. 3.
Fig. 3.

Plot of the resolution degradation factor κ versus the true separation |xx|/λ0 between two in-phase point sources, assuming p=p=p0eiω0t+c.c., T2π/ω0, p0=p0x^, y=y, and z=z. At |xx|=0, the Fisher information matrix is singular [57,58]. κ remains the same for two out-of-phase sources with otherwise the same assumptions.

Equations (71)

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P(Y|X)=tr[E(Y)U(X,T)|ψψ|U(X,T)],
Σμν(X)dYP(Y|X)[X˜μ(Y)Xμ][X˜ν(Y)Xν].
Σ(X)j1(X),
jμν(X)dYP(Y|X)[θμlnP(Y|X)][θνlnP(Y|X)].
Σ(X)j1(X)J1(X),
Jμν(X)=4ReΔgμ(X)Δgν(X),
gμ(X)10TdtU(X,t)H(X,t)XμU(X,t)
Jab(θ)=μ,vXμθaJμ,v(X)Xvθb|X=X(θ).
H(r,t)=HF+HI(r,t),
HF=sd3kω(k)a(k,s)a(k,s),
HI(r,t)=p(t)·E(r),
E(r)=sd3kω2(2π)3ϵ0[iε(k,s)a(k,s)eik·r+H.c.],
a(k,s,t)U(X,t)a(k,s)U(X,t),
=eiωt[a(k,s)+α(k,s,r,t)],
α(k,s,r,T)=ω02(2π)3ϵ0eik·rε*(k,s)·[p0ei(ωω0)T/2sin(ωω0)T/2(ωω0)/2+p0*ei(ω+ω0)T/2sin(ω+ω0)T/2(ω+ω0)/2],
Nsd3k|α(k,s,r,T)|2|p0|2ω03T3πϵ0c3.
Δgμ(r)=2WμΔPμ(r),μ{x,y,z},
ΔPμ(r)12i[Δbμ(r)Δbμ(r)],
Δbμ(r)Wμsd3k[ikμα*(k,s,r,T)]Δa(k,s),
Wμ[sd3kkμ2|αμ(k,s,r,T)|2]1/2.
Wx=Wy=52λ02πN,Wz=5λ02πN,
Wx=Wy=103λ02πN,Wz=52λ02πN,
Jμν(r)=8Wμ2ΔPμ(r)ΔPν(r).
Jμν(r)=4Wμ2δμν,Σμμ(r)Wμ24,
|rr0|λ0,
ΔQμ(r0)12[Δbμ(r0)+Δbμ(r0)],
Σμμ(r)Wμ22ΔQμ2(r0)
ΔQμ2(r0)f(N0)2,ΔPμ2(r0)12f(N0),
f(N0)(2N0+1)[11(2N0+1)2],
f(0)=1,f(N0)14N0forN01.
Σμμ(r)Wμ22ΔQμ2(r0)=Wμ24f(N0).
Σμμ(r)Jμμ1(r)=Wμ28ΔPμ2(r)Wμ24f(N0).
H(r,r,t)=HF+HI(r,r,t),
HI(r,r,t)=p(t)·E(r)p(t)·E(r).
Jxx(X)=Jxx(X)=4Resd3kkx2α*(k,s,r,T)α(k,s,r,T),
Σxx(X)1Jxx[1κ(X)],
κ(X)Jxx2(X)JxxJxx=(Resd3kkx2α*α)2sd3kkx2|α|2sd3kkx2|α|2,
Σ¯μν(Z)dXdYP(Y|X,Z)PX|Z(X|Z)[X˜μ(Y,Z)Xμ]×[X˜ν(Y,Z)Xν],
Σ¯(Z)J¯1(Z),
J¯(Z)=EX|Z[J(X|Z)]+j(Z),
jμν(Z)dXPX|Z(X|Z)[XμlnPX|Z(X|Z)]×[XνlnPX|Z(X|Z)].
ΠμνEZ[Σ¯(Z)],
Σ¯(Z)dXdYP(Y|X,Z)PX|Z(X|Z)[X˜μ(Y)Xμ]×[X˜ν(Y)Xν],
ΠEZ[J¯1(Z)].
ρ=d2p0PZ(p0)U(X,p0,T)ρ0U(X,p0,T).
J¯μν(p0)=4Wμ2(p0)δμν+jμν=N(p0)Cμλ02δμν+jμν,
ΠμμEp0[14/Wμ2(p0)+jμμ]=Ep0[1N(p0)/(Cμλ02)+jμμ].
Ep0[1N(p0)/(Cμλ02)+jμμ]=Cμλ02N¯0dNexp(NN¯)1N+Cμλ02jμμ,
Cμλ02N¯lnN¯Cμλ02jμμ,N¯Cμλ02jμμ.
J¯xx(p0)=4Wx2(p0)+jxx,
J¯xx(p0)=4Wx2(p0)+jxx,
J¯xx(p0,p0)=EX[Jxx(p0,p0)].
ΠxxE(p0,p0){1J¯xx(p0)[1κ¯(p0,p0)]},
κ¯(p0,p0)J¯xx2(p0,p0)J¯xx(p0)J¯xx(p0).
|Ψ=c|0,csd3kϕ(k,s)a(k,s),
ϕ(k,s)=k,s|Ψ=0|a(k,s)|Ψ
ϕ(k,s)=k,s|g|HI|e|0[(ωω˜0)+i/(2T1)]eiωT,
HI=iω0(μ12σμ12*σ)·A(r),
A(r)=sd3k2(2π)3ωϵ0[a(k,s)ε(k,s)eik·r+h.c.],
T1=3πϵ0c3|μ12|2ω03,
ϕ(k,s)=1i(ωω˜0)+1/(2T1)ω022(2π)3ωϵ0μ12·ε*(k,s)×eik·riωT.
F=|0|c(X)c(X+δX)|0|2,
=|[c(X),c(X+δX)]|2,
1+μ,νδXμδXν{Re[c,2cXμXν]+Im[c,cXμ]Im[c,cXν]},
Re0|ccXμ|0=Re[c,cXμ]=0,
[c,cXμ]=sd3kϕ*(k,s)ϕ(k,s)Xμ=0,
Jμν=4Re[c,2cXμXν],
=4Resd3kϕ*(k,s)2ϕ(k,s)XμXν,
=4Resd3kkμkν(ωω˜0)2+1/(4T12)ω022(2π)3ωϵ0×|μ12·ε*(k,s)|2.
Jμν4Resd3kkμkν2πT1δ(ωω0)ω022(2π)3ωϵ0×|μ12·ε*(k,s)|2.
Jμν=4δμνNWμ2δμνλ02,ΣμμJμμ1=NWμ24λ02,

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