Quantum limits to optical point-source localization

Mankei Tsang

doi:10.1364/OPTICA.2.000646

1. INTRODUCTION

The resolution limit of optical microscopes is one of the most important problems in science and engineering. The Abbe–Rayleigh criterion with respect to the free-space wavelength $λ_{0}$ has been a widely used resolution limit [1], but it is now well known that the criterion is heuristic in the context of microscopy and super-resolution microscopy is possible. An important class of super-resolution microscopy, including stimulated-emission-depletion microscopy [2] and photoactivated-localization microscopy [3], relies on the accurate localization of point sources from far field [4]. The localization accuracy, which represents an important measure of the microscope resolution, is then limited by the statistics of the optical measurement [5–8]. Prior analyses of point-source localization accuracy assumed classical, scalar, and paraxial optics with statistics specific to the measurement methods [5–8]. On a more fundamental level, however, optics is governed by the quantum theory of electromagnetic fields [9], and the existence of more accurate measurement methods [10] and more fundamental quantum limits remains an open question. For example, the superoscillation phenomenon [11] suggests that super-resolution diffraction patterns can be obtained at the expense of signal power; can it be exploited to improve the resolution of microscopes [12–14]?

Using a quantum Cramér–Rao bound (QCRB) [15,16] and the full quantum theory of electromagnetic fields [9], here I derive quantum limits to the accuracy of locating point sources. These quantum resolution limits are more general and fundamental than prior classical analyses in the sense that they apply to any measurement method and take full account of the quantum, nonparaxial, and vectoral nature of photons. To arrive at analytic results, I focus mainly on the cases of one and two monochromatic classical sources and an initial vacuum optical state. The possibility of using squeezed light to further enhance the accuracy of locating one point source will also be discussed. To study partially coherent sources, I model incoherence using the concept of nuisance parameters, which are unknown parameters of no primary interest in the context of statistical inference. Quantum bounds for partially coherent sources are then derived by introducing a new generalized QCRB that accounts for nuisance parameters in a special way.

In quantum optics, there has been a substantial literature on quantum imaging (see, for example, Refs. [15,17–38]), but most of them assume certain quantum optical states without considering how they may be generated by objects relevant to microscopy or consider simply the estimation of mirror displacement. Helstrom’s derivation of the QCRB for one point source [15,17] is the most relevant prior work, although he used the paraxial approximation, did not consider the use of squeezed light, and studied two sources only in the context of binary hypothesis testing [15]. There have also been intriguing claims of super-resolution using the nonclassical photon statistics from single-photon sources [34–37], but their protocols have not been analyzed using statistical inference, so even though their images appear sharper, the accuracies of their methods in estimating object parameters remain unclear. To investigate their claims, I also derive a quantum bound for locating a single-photon source.

2. QUANTUM PARAMETER ESTIMATION

Let the initial quantum state of a system be $| ψ 〉$ . After unitary evolution $U (X, T)$ with respect to Hamiltonian $H (X, t)$ as a function of parameters $X = (X_{1}, X_{2}, \dots)$ , the quantum system is measured with outcome $Y$ . The probability distribution of $Y$ according to Born’s rule can be expressed as [15,16,39]

P (Y | X) = tr [E (Y) U (X, T) | ψ 〉 〈 ψ | U^{†} (X, T)],

where

E (Y)

is the positive operator-valued measure (POVM) that characterizes the quantum measurement and tr denotes the operator trace. Denote the estimator of

X

using

Y

as

\tilde{X} (Y)

. The estimation error matrix is defined as

Σ_{μ ν} (X) \equiv \int d Y P (Y | X) [{\tilde{X}}_{μ} (Y) - X_{μ}] [{\tilde{X}}_{ν} (Y) - X_{ν}] .

For unbiased estimators, the classical Cramér–Rao bound states that [40]

Σ (X) \geq j^{- 1} (X),

which means that

Σ - j^{- 1}

is positive semidefinite.

j (X)

is the Fisher information matrix given by

j_{μ ν} (X) \equiv \int d Y P (Y | X) [\frac{\partial}{\partial θ_{μ}} \ln P (Y | X)] [\frac{\partial}{\partial θ_{ν}} \ln P (Y | X)] .

The bound has been used, for example, in Refs. [7,8], to evaluate the point-source localization accuracy for a microscope.

It turns out that for any POVM and thus any measurement in quantum mechanics, another lower bound exists in the form of the QCRB [15,41]:

Σ (X) \geq j^{- 1} (X) \geq J^{- 1} (X),

which means that

j^{- 1} - J^{- 1}

and

Σ - J^{- 1}

are positive semidefinite.

J

is the quantum Fisher information (QFI) matrix; it can be obtained by expressing the fidelity

{| 〈 ψ | U^{†} (X, T) U (X + δ X, T) | ψ 〉 |}^{2}

in the interaction picture [42] and expanding it to the second order of

δ X

[43,44]. The result is

J_{μ ν} (X) = 4 Re 〈 Δ g_{μ} (X) Δ g_{ν} (X) 〉,

where Re denotes the real part,

〈 A 〉 \equiv 〈 ψ | A | ψ 〉

,

Δ A \equiv A - 〈 A 〉

, and

g_{μ} (X) \equiv \frac{1}{ℏ} \int_{0}^{T} d t U^{†} (X, t) \frac{\partial H (X, t)}{\partial X_{μ}} U (X, t)

is the generator of the parameter shift in the Heisenberg picture. For

M

trials, the QFI is simply multiplied by

M

and at least one component of the QCRB can be attained in an asymptotic

M \to \infty

sense [45]. If one wishes to consider a new set of parameters

θ

related to the original set

X

and

X

can be expressed as a function of

θ

, then the new QFI matrix is simply given by

J_{a b}^{'} (θ) = {\sum_{μ, v} \frac{\partial X_{μ}}{\partial θ_{a}} J_{μ, v} (X) \frac{\partial X_{v}}{\partial θ_{b}} |}_{X = X (θ)} .

Various generalizations of the QCRB and alternatives are available [15,46–49], but the version presented suffices to illustrate the pertinent physics. In Section 6, the QCRB will be generalized to a Bayesian version that treats nuisance parameters separately and is used to study partially coherent sources.

3. ONE CLASSICAL POINT SOURCE

Consider first a classical point source, as depicted in Fig. 1. The Hamiltonian is [9]

H (r, t) = H_{F} + H_{I} (r, t),

H_{F} = \sum_{s} \int d^{3} k ℏ ω (k) a^{†} (k, s) a (k, s),

H_{I} (r, t) = - p (t) \cdot E (r),

E (r) = \sum_{s} \int d^{3} k \sqrt{\frac{ℏ ω}{2 {(2 π)}^{3} ϵ_{0}}} [i ε (k, s) a (k, s) e^{i k \cdot r} + H.c.],

where

k = k_{x} \hat{x} + k_{y} \hat{y} + k_{y} \hat{z}

is a wave vector,

(\hat{x}, \hat{y}, \hat{z})

denote unit vectors in the Cartesian coordinate system,

\int d^{3} k \equiv \int_{- \infty}^{\infty} d k_{x} \int_{- \infty}^{\infty} d k_{y} \int_{- \infty}^{\infty} d k_{z}

,

s

is an index for the two polarizations,

ε (k, s)

is a unit polarization vector,

ω (k) = c | k |

(

c

is the speed of light),

p (t)

is the c-number dipole moment of the source,

r = x \hat{x} + y \hat{y} + z \hat{z}

is its position,

ϵ_{0}

is the free-space permittivity,

a (k, s)

is an annihilation operator obeying the commutation relation

[a (k, s), a^{†} (k^{'}, s^{'})] = δ_{s s^{'}} δ^{3} (k - k^{'})

, and H.c. denotes the Hermitian conjugate. Since

p (t)

is a c-number,

H_{I}

implements a field displacement operation [9]. The Heisenberg picture of

a (k, s)

is

a (k, s, t) \equiv U^{†} (X, t) a (k, s) U (X, t),

= e^{- i ω t} [a (k, s) + α (k, s, r, t)],

where

α

is the radiated field. Assuming

p (t) = p_{0} e^{- i ω_{0} t} + c.c.

, where c.c. denotes the complex conjugate,

α

becomes

α (k, s, r, T) = \sqrt{\frac{ω_{0}}{2 {(2 π)}^{3} ℏ ϵ_{0}}} e^{- i k \cdot r} ε^{*} (k, s) \cdot [p_{0} e^{i (ω - ω_{0}) T / 2} \frac{\sin (ω - ω_{0}) T / 2}{(ω - ω_{0}) / 2} + p_{0}^{*} e^{i (ω + ω_{0}) T / 2} \frac{\sin (ω + ω_{0}) T / 2}{(ω + ω_{0}) / 2}],

which indicates that only optical modes with

ω (k) = ω_{0}

grow in time, corresponding to the far field, whereas all the other near-field modes with

ω (k) \neq ω_{0}

oscillate. The

ω (k) = ω_{0}

relation specifies the spatial frequencies available to the far optical fields [50,51]. Assuming

T ≫ 2 π / ω_{0}

, such that

\sin^{2} [(ω \pm ω_{0}) T / 2] / {[(ω \pm ω_{0}) / 2]}^{2} \approx 2 π T δ (ω \pm ω_{0})

, using the identity

\sum_{s} ε_{μ} (k, s) ε_{ν}^{*} (k, s) = δ_{μ ν} - k_{μ} k_{ν} / {| k |}^{2}

[9], and switching to the spherical coordinate system for

k

, it can be shown that the average number of radiated photons for an initial vacuum state is

N \equiv \sum_{s} \int d^{3} k {| α (k, s, r, T) |}^{2} \approx \frac{{| p_{0} |}^{2} ω_{0}^{3} T}{3 π ℏ ϵ_{0} c^{3}} .

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.

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The far-field limit ( $ω_{0} T \to \infty$ ) will be assumed hereafter.

I now focus on two representative cases: a linearly polarized dipole with $p_{0} = p_{0} \hat{z}$ and a circularly polarized dipole with $p_{0} = p_{0} (\hat{x} + i \hat{y}) / \sqrt{2}$ . Taking the unknown parameters to be $r$ , the generators for $X = (x, y, z)$ can be expressed, after some algebra, as

Δ g_{μ} (r) = - \frac{\sqrt{2}}{W_{μ}} Δ P_{μ} (r), μ \in {x, y, z},

Δ P_{μ} (r) \equiv \frac{1}{\sqrt{2} i} [Δ b_{μ} (r) - Δ b_{μ}^{†} (r)],

where

Δ b_{μ}

is a normalized annihilation operator defined as

Δ b_{μ} (r) \equiv W_{μ} \sum_{s} \int d^{3} k [- i k_{μ} α^{*} (k, s, r, T)] Δ a (k, s),

such that

[Δ b_{μ} (r), Δ b_{ν}^{†} (r)] = δ_{μ ν}

, and the normalization constants

W_{μ}

are

W_{μ} \equiv {[\sum_{s} \int d^{3} k k_{μ}^{2} {| α_{μ} (k, s, r, T) |}^{2}]}^{- 1 / 2} .

The

d^{3} k

integrals can again be computed with the help of the far-field assumption and spherical coordinates. The results depend on

p_{0}

; for

p_{0} = p_{0} \hat{z}

,

W_{x} = W_{y} = \sqrt{\frac{5}{2}} \frac{λ_{0}}{2 π \sqrt{N}}, W_{z} = \frac{\sqrt{5} λ_{0}}{2 π \sqrt{N}},

and for

p_{0} = p_{0} (\hat{x} + i \hat{y}) / \sqrt{2}

,

W_{x} = W_{y} = \sqrt{\frac{10}{3}} \frac{λ_{0}}{2 π \sqrt{N}}, W_{z} = \sqrt{\frac{5}{2}} \frac{λ_{0}}{2 π \sqrt{N}},

but the important point here is that they are all of the order of

λ_{0} / \sqrt{N}

, where

λ_{0} \equiv 2 π c / ω_{0}

is the free-space wavelength. The QFI becomes

J_{μ ν} (r) = \frac{8}{W_{μ}^{2}} 〈 Δ P_{μ} (r) Δ P_{ν} (r) 〉 .

For an initial vacuum state (or any coherent state),

〈 Δ P_{μ} (r) Δ P_{ν} (r) 〉 = δ_{μ ν} / 2

and the QCRB is hence

J_{μ ν} (r) = \frac{4}{W_{μ}^{2}} δ_{μ ν}, Σ_{μ μ} (r) \geq \frac{W_{μ}^{2}}{4},

meaning that the quantum resolution limit in terms of the root-mean-square error

\sqrt{Σ_{μ μ}}

is of the order of

λ_{0} / \sqrt{N}

. I call this limit the quantum shot-noise limit. Generalization to lossless media is straightforward and results simply in

λ_{0}

being replaced by the wavelength in the medium. Section 9 shows that a single-photon source also obeys this limit with repeated trials.

Assuming uncorrelated photons, Refs. [5–7,25] derived a similar limit in the form of $σ / \sqrt{N}$ , where $σ$ is the width of the imaging point-spread function. While their limit also scales as $1 / \sqrt{N}$ , all those analyses assume the paraxial approximation and measurement by a photon-counting camera, whereas the quantum shot-noise limit here is valid for any numerical aperture and any measurement, including common methods such as photon counting, homodyne/heterodyne detection, and digital holography. The limit in Refs. [5–7,25] also implies that the quantum shot-noise limit is reasonably tight, as the camera measurement with suitable postprocessing can at least follow the quantum-optimal shot-noise scaling. Section 4 shows that homodyne measurement with a special local-oscillator field can also approach the quantum limit if the radiation is coherent.

For a concrete numerical example, consider the semiclassical paraxial analysis of conventional single-molecule microscopy by Ober et al. [7] who used the classical Cramér–Rao bound and a shot-noise assumption to derive a limit of 2.301 nm on the root-mean-square localization error for a free-space wavelength of 520 nm, a numerical aperture of 1.4, a photon collection efficiency of 0.033, a photon flux of $2 \times 10^{6} s^{- 1}$ , and an acquisition time of 0.01 s. If the efficiency were 1, their limit would become 0.418 nm. In comparison, if I take the refractive index of the immersion oil to be 1.52, $λ_{0} = 520 nm / 1.52 = 342 nm$ to be the wavelength in the medium, and the photon number to be $N = 2 \times 10^{4}$ , then the quantum shot-noise limit according to Eqs. (21), (22), and (24) is $λ_{0} / (2 π \sqrt{N}) = 0.385 nm$ times a constant factor close to 1.

It remains to be seen whether superoscillation techniques are similarly efficient, but the key point here is that since the quantum bound is valid for any measurement and conventional methods can already get close to it, no other measurement technique is able to offer any significant advantage in resolution enhancement over the conventional methods.

4. QUANTUM ENHANCEMENT

Even though the source is classical, quantum enhancement is possible if the initial state $| ψ 〉$ is nonclassical, as I now show. If $Δ g_{μ}$ were independent of the parameter, the accuracy could be enhanced by squeezing and measuring the conjugate quadrature [52]. Although $Δ g_{μ} (r)$ depends on the unknown $r$ here, the radiated field can be approximated as $α (k, s, r, T) \approx α (k, s, r_{0}, T)$ , resulting in $Δ g_{μ} (r) \approx Δ g_{μ} (r_{0})$ , provided

| r - r_{0} | ≪ λ_{0},

with respect to a known reference position

r_{0}

. The acquisition of such prior information will require a fixed amount of overhead resource; but once it is done, one can squeeze the quadrature,

Δ Q_{μ} (r_{0}) \equiv \frac{1}{\sqrt{2}} [Δ b_{μ} (r_{0}) + Δ b_{μ}^{†} (r_{0})],

in the initial state and perform a homodyne measurement of

Δ Q_{μ} (r_{0})

to estimate

r

much more accurately. Since

[Δ Q_{μ} (r_{0}), Δ Q_{ν} (r_{0})] = 0

, all three quadratures can be squeezed and measured simultaneously in principle. The estimation error becomes

Σ_{μ μ} (r) \approx \frac{W_{μ}^{2}}{2} 〈 Δ Q_{μ}^{2} (r_{0}) 〉

and the error reduction below the shot-noise limit is determined by the squeezing factor, which is limited by the average photon number

N_{0}

in the initial state (not to be confused with

N

). Using

〈 Δ Q_{μ}^{2} (r_{0}) 〉 + 〈 Δ P_{μ}^{2} (r_{0}) 〉 \leq 2 N_{0} + 1

and the uncertainty relation

〈 Δ Q_{μ}^{2} 〉 〈 Δ P_{μ}^{2} 〉 \geq 1 / 4

, it can be shown that

〈 Δ Q_{μ}^{2} (r_{0}) 〉 \geq \frac{f (N_{0})}{2}, 〈 Δ P_{μ}^{2} (r_{0}) 〉 \leq \frac{1}{2 f (N_{0})},

f (N_{0}) \equiv (2 N_{0} + 1) [1 - \sqrt{1 - {(2 N_{0} + 1)}^{- 2}}],

where

f (0) = 1, f (N_{0}) \approx \frac{1}{4 N_{0}} for N_{0} ≫ 1 .

With a zero-mean minimum-uncertainty state and all initial photons in the

Δ b_{μ} (r_{0})

mode, the estimation error becomes

Σ_{μ μ} (r) \approx \frac{W_{μ}^{2}}{2} 〈 Δ Q_{μ}^{2} (r_{0}) 〉 = \frac{W_{μ}^{2}}{4} f (N_{0}) .

The enhancement factor

f (N_{0})

is optimal as the QCRB can be further bounded by

Σ_{μ μ} (r) \geq J_{μ μ}^{- 1} (r) = \frac{W_{μ}^{2}}{8 〈 Δ P_{μ}^{2} (r) 〉} \geq \frac{W_{μ}^{2}}{4} f (N_{0}) .

This means that squeezed light with average photon number

N_{0}

can beat the quantum shot-noise limit to the mean-square error by roughly a factor of

N_{0}

.

The optical mode to be squeezed has a profile $i k_{μ} α (k, s, r_{0}, T)$ . This means that, in real space, the electric field profile of the mode should be the spatial derivative of the radiated field. This kind of squeezing and measurement has actually been demonstrated experimentally, albeit in the paraxial regime, by Taylor et al. in the context of particle tracking [38] where the weak scatterer under a strong pump can be modeled as a classical source, similar to the implementation of field displacement by a beam splitter [53], and the spatial mode profile of the squeezed light and the local oscillator is a spatial derivative of the scattered field. To realize an enhancement in practice, accurate phase locking of the squeezed light and the local oscillator to the radiated field and a high measurement efficiency are crucial. Phase locking cannot be achieved with incoherent point sources such as fluorescent markers, but can be done with dielectric particles via Rayleigh scattering [38] or second-harmonic nanoparticles [54,55]; the latter are especially promising for biological imaging applications [56].

5. TWO CLASSICAL POINT SOURCES

Next, consider two classical point sources at $r$ and $r^{'}$ , as shown in Fig. 2. The Hamiltonian is now

H (r, r^{'}, t) = H_{F} + H_{I} (r, r^{'}, t),

H_{I} (r, r^{'}, t) = - p (t) \cdot E (r) - p^{'} (t) \cdot E (r^{'}) .

Fig. 2. Two classical point sources with dipole moments $p (t)$ and $p^{'} (t)$ at $r$ and $r^{'}$ , respectively, with quantum optical radiation.

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The Heisenberg picture of $a (k, s)$ becomes $a (k, s, t) = e^{- i ω t} [a (k, s) + α (k, s, r, t) + α^{'} (k, s, r^{'}, t)]$ , where $α$ and $α^{'}$ are the radiated fields from the two sources, $α$ is the same as before, and $α^{'}$ has the same expression as $α$ , except that $p$ is replaced by $p^{'}$ and $r$ by $r^{'}$ . One can then follow the preceding procedure to obtain the QCRB for estimating $r$ and $r^{'}$ . To highlight the important physics, however, consider here the estimation of just two parameters $X = (x, x^{'})$ . The generators $Δ g_{x}$ and $Δ g_{x^{'}}$ may not commute and the QFI matrix for an initial vacuum or any coherent state now has off-diagonal components,

J_{x x^{'}} (X) = J_{x^{'} x} (X) = 4 Re \sum_{s} \int d^{3} k k_{x}^{2} α^{*} (k, s, r, T) α^{'} (k, s, r^{'}, T),

whereas

J_{x x}

remains the same and

J_{x^{'} x^{'}}

has a similar expression to

J_{x x}

.

J_{x x}

and

J_{x^{'} x^{'}}

still obey a shot-noise scaling with the average photon number, but the nonzero off-diagonal components mean that the parameters act as nuisance parameters to each other and the QCRB with respect to, say,

x

is always raised:

Σ_{x x} (X) \geq \frac{1}{J_{x x} [1 - κ (X)]},

where the resolution degradation factor, defined as

κ (X) \equiv \frac{J_{x x^{'}}^{2} (X)}{J_{x x} J_{x^{'} x^{'}}} = \frac{{(Re \sum_{s} \int d^{3} k k_{x}^{2} α^{*} α^{'})}^{2}}{\sum_{s} \int d^{3} k k_{x}^{2} {| α |}^{2} \sum_{s} \int d^{3} k k_{x}^{2} {| α^{'} |}^{2}},

is within the range

0 \leq κ \leq 1

and is determined by the overlap between the two differential mode profiles. The nuisance-parameter effect generalizes the Rayleigh criterion and other classical results [8] by revealing a fundamental measurement-independent degradation of resolution for two point sources with overlapping radiation. For example, Fig. 3 plots

κ

against

| x - x^{'} | / λ_{0}

, assuming

p = p^{'} = p_{0} e^{- i ω_{0} t} + c.c.

,

T ≫ 2 π / ω_{0}

,

p_{0} = p_{0} \hat{x}

,

y = y^{'}

, and

z = z^{'}

.

κ \approx 0

for

| x - x^{'} | ≫ λ_{0}

, as expected, but it approaches 1 and leads to a diverging QCRB when

| x - x^{'} | ≪ λ_{0}

. Section 8 shows that the degradation effect should still exist for two partially coherent sources.

Fig. 3. Plot of the resolution degradation factor $κ$ versus the true separation $| x - x^{'} | / λ_{0}$ between two in-phase point sources, assuming $p = p^{'} = p_{0} e^{- i ω_{0} t} + c.c.$ , $T ≫ 2 π / ω_{0}$ , $p_{0} = p_{0} \hat{x}$ , $y = y^{'}$ , and $z = z^{'}$ . At $| x - x^{'} | = 0$ , the Fisher information matrix is singular [57,58]. $κ$ remains the same for two out-of-phase sources with otherwise the same assumptions.

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The degradation effect can be avoided by minimizing the overlap before each source is located independently. The overlap can be reduced by making the radiated fields separate in space, time, frequency, quadrature, or polarization; the time multiplexing of point sources has especially been the key driver in current super-resolution microscopy [2–4].

Note that $κ$ also depends on the relative phase between $α$ and $α^{'}$ . For example, under the assumptions in the caption of Fig. 3, it can be shown that the QFI matrix transformed with respect to the average position $(x + x^{'}) / 2$ and the separation $x - x^{'}$ is diagonal. The QFI component with respect to the average position still obeys a shot-noise scaling, while the increase in $κ$ can be traced to the increased error in estimating the separation. Similarly, when $α$ and $α^{'}$ are 180° out of phase but otherwise obey the same assumptions, $κ$ remains the same but its increase is now due to the increased error for the average position. An interesting scenario occurs when $α$ and $α^{'}$ are 90° out of phase: the two fields are in orthogonal quadratures, $κ$ becomes 0, and they can be measured separately using homodyne or heterodyne detection. This phenomenon suggests that structured illumination [51,59–61] can be used to excite the sources, such that their relative phase and amplitude can be controlled to some degree and the overlap can be reduced for certain ranges of parameters.

When the overlap is unavoidable or when the generators do not commute, heterodyne measurements can still be used to measure both quadratures of $a (k, s, T)$ and should have a classical Fisher information within a factor of 1/2 of the QFI. Quantum enhancement may also be possible using entangled squeezed states [62]; the specific experimental design will be left for future studies.

6. BAYESIAN QCRB WITH NUISANCE PARAMETERS

Incoherent sources are characterized by the statistical fluctuations of the fields [9]. For point-source radiation, incoherence originates from the randomness of the amplitude, phase, and direction of the point dipole. In the context of statistical inference, these random variables are most suitably modeled as nuisance parameters. There are many ways to generalize the Cramér–Rao bounds when nuisance parameters are present [63]. The previous sections show one way, which includes the nuisance parameters as part of the wanted parameters $X$ . To derive tighter bounds for other types of nuisance parameters, here I start with a Bayesian QCRB and generalize a classical approach by Miller and Chang [63,64]. Let $Z$ be a set of nuisance parameters and suppose first that $Z$ is given. The Bayesian estimation error matrix is

{\bar{Σ}}_{μ ν} (Z) \equiv \int d X d Y P (Y | X, Z) P_{X | Z} (X | Z) [{\tilde{X}}_{μ} (Y, Z) - X_{μ}] \times [{\tilde{X}}_{ν} (Y, Z) - X_{ν}],

where

P_{X | Z} (X | Z)

is the prior distribution of

X

conditioned on

Z

. Note that this Bayesian definition of error regards

X

as a random parameter by averaging over its prior and is different from the frequentist definition in Eq. (2) [40]. A Bayesian QCRB valid for any estimator is given by [46,47]

\bar{Σ} (Z) \geq {\bar{J}}^{- 1} (Z),

\bar{J} (Z) = E_{X | Z} [J (X | Z)] + j (Z),

where

J (X | Z)

is the same QFI as before, except that it is now conditioned on

Z

,

E_{X | Z}

denotes expectation over

P_{X | Z} (X | Z)

, and

j (Z)

is a prior Fisher information defined as

j_{μ ν} (Z) \equiv \int d X P_{X | Z} (X | Z) [\frac{\partial}{\partial X_{μ}} \ln P_{X | Z} (X | Z)] \times [\frac{\partial}{\partial X_{ν}} \ln P_{X | Z} (X | Z)] .

If

Z

is a random parameter with prior distribution given by

P_{Z} (Z)

, the estimation error is

Π_{μ ν} \equiv E_{Z} [{\bar{Σ}}^{'} (Z)],

{\bar{Σ}}^{'} (Z) \equiv \int d X d Y P (Y | X, Z) P_{X | Z} (X | Z) [{\tilde{X}}_{μ} (Y) - X_{μ}] \times [{\tilde{X}}_{ν} (Y) - X_{ν}],

where

E_{Z}

denotes expectation over

P_{Z}

and the estimator

{\tilde{X}}_{μ} (Y)

can no longer depend on

Z

. The lower bound in Eq. (39) still holds for

{\bar{Σ}}^{'} (Z)

, so one can obtain a lower bound on

Π

given by

Π \geq E_{Z} [{\bar{J}}^{- 1} (Z)] .

The important mathematical feature of this bound is that the expectation with respect to the nuisance parameter

Z

is taken after the inverse of the conditional QFI matrix. This can sometimes lead to a tighter bound than a QCRB that includes

Z

as part of

X

. Note also that this Bayesian bound is valid for any estimator, not just the unbiased ones, unlike the claim in Ref. [64]. The tightness of the bound should depend on whether the nuisance parameters can be accurately estimated from the measurements.

7. ONE PARTIALLY COHERENT SOURCE

I now use the new bound to study partially coherent sources. First, consider the example of one point source in Section 3, but suppose that the complex dipole amplitude $p_{0}$ is unknown. Assuming $Z = p_{0}$ , the quantum state before measurement is

ρ = \int d^{2} p_{0} P_{Z} (p_{0}) U (X, p_{0}, T) ρ_{0} U^{†} (X, p_{0}, T) .

If

ρ_{0}

is a vacuum state,

U ρ_{0} U^{†}

is a coherent state and

ρ

is a classical mixed state of light with

P_{Z} (p_{0})

determining the Sudarshan–Glauber

P

function [9]. The random

p_{0}

, therefore, gives rise to a classical partially coherent source model. For an initial vacuum,

\bar{J} (p_{0})

is given by

{\bar{J}}_{μ ν} (p_{0}) = \frac{4}{W_{μ}^{2} (p_{0})} δ_{μ ν} + j_{μ ν} = \frac{N (p_{0})}{C_{μ} λ_{0}^{2}} δ_{μ ν} + j_{μ ν},

where

W_{μ}

and

N

now depend on the unknown dipole moment and

C_{μ}

is a constant of the order of 1 that can be determined from Eqs. (21) or (22). Assuming that

j

is diagonal and independent of

p_{0}

and taking the inverse and then the expectation according to Eq. (44), I obtain

Π_{μ μ} \geq E_{p_{0}} [\frac{1}{4 / W_{μ}^{2} (p_{0}) + j_{μ μ}}] = E_{p_{0}} [\frac{1}{N (p_{0}) / (C_{μ} λ_{0}^{2}) + j_{μ μ}}] .

For example, if

P_{Z} (p_{0})

corresponds to the

P

function of a thermal source, then the bound can be written in terms of the average radiated photon number

\bar{N}

as

E_{p_{0}} [\frac{1}{N (p_{0}) / (C_{μ} λ_{0}^{2}) + j_{μ μ}}] = \frac{C_{μ} λ_{0}^{2}}{\bar{N}} \int_{0}^{\infty} d N \exp (- \frac{N}{\bar{N}}) \frac{1}{N + C_{μ} λ_{0}^{2} j_{μ μ}},

\approx \frac{C_{μ} λ_{0}^{2}}{\bar{N}} \ln \frac{\bar{N}}{C_{μ} λ_{0}^{2} j_{μ μ}}, \frac{\bar{N}}{C_{μ} λ_{0}^{2}} ≫ j_{μ μ} .

An alternative Bayesian QCRB can be obtained by including

p_{0}

as part of

X

. In that case, the off-diagonal QFI elements between

p_{0}

and

r

turn out to be zero, and the expectation with respect to

p_{0}

is taken before the inverse, leading to a bound given by

C_{μ} λ_{0}^{2} / \bar{N}

. The separate treatment of

p_{0}

as a nuisance parameter here involves taking the expectation after the inverse, giving rise to an additional factor

\sim \ln \bar{N}

and, thus, a tighter bound for a large

\bar{N}

. In general, Jensen’s inequality can be used to show that the shot-noise scaling with respect to the average photon number

E_{p_{0}} [N (p_{0})]

cannot be beat for any non-negative P function.

8. TWO PARTIALLY COHERENT SOURCES

Next, consider the example of two point sources in Section 5, and $Z = (p_{0}, p_{0}^{'})$ is now assumed to be unknown to model partially coherent sources. Assuming again that $j$ is diagonal and $X = (x, x^{'})$ is independent of $Z$ ,

{\bar{J}}_{x x} (p_{0}) = \frac{4}{W_{x}^{2} (p_{0})} + j_{x x},

{\bar{J}}_{x^{'} x^{'}} (p_{0}^{'}) = \frac{4}{W_{x^{'}}^{2} (p_{0}^{'})} + j_{x^{'} x^{'}},

{\bar{J}}_{x x^{'}} (p_{0}, p_{0}^{'}) = E_{X} [J_{x x^{'}} (p_{0}, p_{0}^{'})] .

{\bar{J}}_{x x^{'}}

is now the expectation of

J_{x x^{'}}

over

X = (x, x^{'})

, conditioned on the dipole moments. If the two point sources are a priori known to be close relative to

λ_{0}

, then

{\bar{J}}_{x x^{'}} (p_{0}, p_{0}^{'})

can still have a significant magnitude for certain

(p_{0}, p_{0}^{'})

. The bound given by Eq. (44) becomes

Π_{x x} \geq E_{(p_{0}, p_{0}^{'})} {\frac{1}{{\bar{J}}_{x x} (p_{0}) [1 - \bar{κ} (p_{0}, p_{0}^{'})]}},

where a new resolution degradation factor is defined as

\bar{κ} (p_{0}, p_{0}^{'}) \equiv \frac{{\bar{J}}_{x x^{'}}^{2} (p_{0}, p_{0}^{'})}{{\bar{J}}_{x x} (p_{0}) {\bar{J}}_{x^{'} x^{'}} (p_{0}^{'})} .

The important point here is that

\bar{κ} (p_{0}, p_{0}^{'})

can still be close to 1 for certain values of

(p_{0}, p_{0}^{'})

if the two sources are a priori known to be close to each other relative to

λ_{0}

,

1 / [1 - \bar{κ} (p_{0}, p_{0}^{'})] ≫ 1

is possible, and the expectation over

(p_{0}, p_{0}^{'})

will then be dominated by those large values. In other words, the resolution degradation effect derived for coherent sources must still exist for partially coherent sources if their radiated fields may have significant overlap.

An alternative Bayesian QCRB that includes $(p_{0}, p_{0}^{'})$ as part of $X$ can again be computed, but at some stage it involves taking the expectation of ${\bar{J}}_{x x^{'}}$ with respect to $(p_{0}, p_{0}^{'})$ before the inverse is taken. For incoherent sources, this can reduce the off-diagonal components significantly; the resulting bound, while still valid, would be less tight and no longer demonstrate the resolution degradation effect.

9. SINGLE-PHOTON SOURCE

Consider now an initially excited two-level atom in free space. A detailed analysis of atom–photon interaction is formidable [9,65], but when the initial optical state is vacuum, spontaneous emission can be treated more easily, as the atom must decay to ground state in the long-time limit and the final optical state must contain exactly one photon. Using the continuous Fock space [9], the final optical state in the Schrödinger picture can be written with respect to the vacuum $| 0 〉$ as

| Ψ 〉 = c^{†} | 0 〉, c^{†} \equiv \sum_{s} \int d^{3} k ϕ (k, s) a^{†} (k, s),

where

ϕ (k, s) = 〈 k, s | Ψ 〉 = 〈 0 | a (k, s) | Ψ 〉

is the one-photon configuration-space amplitude. Following Chap. III.C of Ref. [65], it can be expressed as

ϕ (k, s) = \frac{〈 k, s | \otimes 〈 g | H_{I} | e 〉 \otimes | 0 〉}{ℏ [(ω - {\tilde{ω}}_{0}) + i / (2 T_{1})]} e^{- i ω T},

H_{I} = i ω_{0} (μ_{12} σ - μ_{12}^{*} σ^{†}) \cdot A (r),

A (r) = \sum_{s} \int d^{3} k \sqrt{\frac{ℏ}{2 {(2 π)}^{3} ω ϵ_{0}}} [a (k, s) ε (k, s) e^{i k \cdot r} + h.c.],

T_{1} = \frac{3 π ℏ ϵ_{0} c^{3}}{{| μ_{12} |}^{2} ω_{0}^{3}},

where

| e 〉

and

| g 〉

are the excited and ground atomic states, respectively,

ω_{0}

is the atomic resonance frequency,

{\tilde{ω}}_{0}

is the Lamb-shifted atomic frequency,

T_{1}

is the decay time,

μ_{12}

is the off-diagonal atomic dipole moment, and

σ \equiv | g 〉 〈 e |

is the atomic lowering operator. The result is

ϕ (k, s) = \frac{1}{i (ω - {\tilde{ω}}_{0}) + 1 / (2 T_{1})} \sqrt{\frac{ω_{0}^{2}}{2 {(2 π)}^{3} ℏ ω ϵ_{0}}} μ_{12} \cdot ε^{*} (k, s) \times e^{- i k \cdot r - i ω T} .

Consider the fidelity

F = {| 〈 0 | c (X) c^{†} (X + δ X) | 0 〉 |}^{2},

= {| [c (X), c^{†} (X + δ X)] |}^{2},

\approx 1 + \sum_{μ, ν} δ X_{μ} δ X_{ν} {Re [c, \frac{\partial^{2} c^{†}}{\partial X_{μ} \partial X_{ν}}] + Im [c, \frac{\partial c^{†}}{\partial X_{μ}}] Im [c, \frac{\partial c^{†}}{\partial X_{ν}}]},

where the fact

Re 〈 0 | c \frac{\partial c^{†}}{\partial X_{μ}} | 0 〉 = Re [c, \frac{\partial c^{†}}{\partial X_{μ}}] = 0,

due to

〈 0 | c (X + δ X) c^{†} (X + δ X) | 0 〉 = 〈 0 | c (X) c^{†} (X) | 0 〉 = 1

, has been used. It can further be shown that

[c, \frac{\partial c^{†}}{\partial X_{μ}}] = \sum_{s} \int d^{3} k ϕ^{*} (k, s) \frac{\partial ϕ (k, s)}{\partial X_{μ}} = 0,

leading to a QFI given by

J_{μ ν} = - 4 Re [c, \frac{\partial^{2} c^{†}}{\partial X_{μ} \partial X_{ν}}],

= - 4 Re \sum_{s} \int d^{3} k ϕ^{*} (k, s) \frac{\partial^{2} ϕ (k, s)}{\partial X_{μ} \partial X_{ν}},

= 4 Re \sum_{s} \int d^{3} k \frac{k_{μ} k_{ν}}{{(ω - {\tilde{ω}}_{0})}^{2} + 1 / (4 T_{1}^{2})} \frac{ω_{0}^{2}}{2 {(2 π)}^{3} ℏ ω ϵ_{0}} \times {| μ_{12} \cdot ε^{*} (k, s) |}^{2} .

Assuming that the decay time is much longer than the optical period (

T_{1} ≫ 2 π / {\tilde{ω}}_{0})

and the Lamb shift is much smaller than the optical

ω_{0}

(

{\tilde{ω}}_{0} \approx ω_{0}

), the QFI becomes

J_{μ ν} \approx 4 Re \sum_{s} \int d^{3} k k_{μ} k_{ν} 2 π T_{1} δ (ω - ω_{0}) \frac{ω_{0}^{2}}{2 {(2 π)}^{3} ℏ ω ϵ_{0}} \times {| μ_{12} \cdot ε^{*} (k, s) |}^{2} .

This turns out to be identical to the QFI derived in Section 3 for an

N = 1

classical source:

J_{μ ν} = \frac{4 δ_{μ ν}}{N W_{μ}^{2}} \sim \frac{δ_{μ ν}}{λ_{0}^{2}}, Σ_{μ μ} \geq J_{μ μ}^{- 1} = \frac{N W_{μ}^{2}}{4} \sim λ_{0}^{2},

where

N

is defined in Eq. (16) and

W_{μ}

is defined in Eq. (20) and given by Eqs. (21) or Eqs. (22), such that

N W_{μ}^{2}

is of the order of

λ_{0}^{2}

. This result shows that a single-photon source offers no fundamental advantage over a classical source that emits one photon on average. Super-resolution beyond the classical Abbe–Rayleigh limit can still be obtained, however, if the experiment can be repeated. The QFI is then multiplied by the number of trials

M

, which is also the total number of emitted photons, and the resulting QCRB is identical to that for a classical source with

M

replacing

N

. The experiments reported in Refs. [35–37] certainly involved a large number of measurements of many photons in total, which can explain the apparent super-resolution, but it remains to be seen whether their methods are accurate or efficient in estimating object parameters.

For two atoms with a large separation ( $| r - r^{'} | ≫ λ_{0}$ ), the one-atom analysis is expected to be applicable to each atom independently. The analysis of two close atoms is much more challenging because of atomic cooperative effects such as Dicke super-radiance [9] and Förster resonance energy transfer [4]. Beyond the current assumption of spontaneous emission, it will also be interesting, although highly nontrivial, to analyze the interaction between two-level atoms and other states of light, such as coherent states or squeezed states, and investigate their quantum localization limits and the possibility of quantum enhancement.

10. CONCLUSION

I have derived quantum limits to point-source localization using quantum estimation theory and the quantum theory of electromagnetic fields. These results not only provide general no-go theorems concerning microscope resolution, but should also motivate further progress in microscopy through classical or quantum techniques beyond the current assumptions. For example, the theory presented may be applied to other more exotic quantum states of light interacting with quantum sources, such as multilevel atoms [9,66], quantum dots [35,67], diamond defects [36,37,68], and second-harmonic nanoparticles [54–56]. It is also possible to generalize the current formalism for open quantum systems [69–72] to account for mixed states, decoherence, optical losses, background noises, and imperfect measurement efficiency. The phenomenon of fluorescence intermittency or blinking represents another interesting challenge to the statistical analysis, as a fluorescent source can turn itself on and off randomly and the localization of two blinking sources offers another route to super-resolution [73]. If done with care, the application of quantum information science to microscopy is destined to yield sound insights and opportunities in both fields.

Funding

National Research Foundation-Prime Minister’s office, Republic of Singapore (NRF) (NRF-NRFF2011-07).

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Quantum limits to optical point-source localization

Abstract

1. INTRODUCTION

2. QUANTUM PARAMETER ESTIMATION

3. ONE CLASSICAL POINT SOURCE

4. QUANTUM ENHANCEMENT

5. TWO CLASSICAL POINT SOURCES

6. BAYESIAN QCRB WITH NUISANCE PARAMETERS

7. ONE PARTIALLY COHERENT SOURCE

8. TWO PARTIALLY COHERENT SOURCES

9. SINGLE-PHOTON SOURCE

10. CONCLUSION

Funding

REFERENCES

Cited By

Figures (3)

Equations (71)

Optica