Quantum imaging with sub-Poissonian light: challenges and perspectives in optical metrology

The DL, $R\simeq0.61 \lambda/NA$ (with $\lambda$ being the wavelength of the light and NA the numerical aperture of the imaging system), represents the maximum obtainable imaging resolution in classical far-field imaging/microscopy. Depending on the kind of light radiation involved, namely incoherent or coherent, it takes the name of Abbe or Rayleigh diffraction limit, respectively.

The DL provides a lower bound to the current capability of precisely measuring the position of objects, from the small one such as e.g. single-photon emitters (colour centres, quantum dots, etc) [20–24], to distant stars. In general, the research of methods to obtain imaging and microscopy resolution below the diffraction limit is a topic of the utmost interest [25–31] that could provide dramatic improvement in the observation of several systems spanning from quantum dots to living cells [5, 32–35], to distant astronomical objects. As a notable example, in several entanglement-related experiments using strongly coupled single-photon emitters it is of the utmost importance to measure their positions with the highest spatial resolution. In principle, this limitation is overcome in microscopy by recently developed techniques such as e.g. stimulated emission depletion (STED) and ground state depletion (GSD) [36, 37], leading their inventors to win the Nobel Prize in 2014. Nevertheless, even if they have been demonstrated to effectively provide super-resolved imaging in a lot of specific applications, including colour centres in diamond [38], they are characterised by rather specific experimental requirements (dual laser excitation system, availability of luminescence quenching mechanisms by stimulated emission, non-trivial shaping of the quenching beam, high power).

In this paper we will focus more on proposal and experiments of sub-diffraction imaging techniques exploiting the peculiar properties of quantum light, rather the ones connected to structured light and individually addressed or quenched fluorophores.

The sensitivity bound is established by the laws of quantum mechanics [39–42] and is related to the mean energy of the probe beam. In particular, in standard imaging and sensing exploiting classical probes, the sensitivity is fundamentally lower bounded by the SNL, $U_{\rm SNL}\sim\langle N_{\rm P} \rangle^{-1/2}$, where $\langle N_{\rm P} \rangle$ is the mean number of photons in the probe.

Since the definition of classicality is not univocal, here we intend as 'classical states' those providing experimental outcomes that would be completely explainable within the semiclassical theory of photodetection, in which the light is treated as a electromagnetic wave and the photocurrent is discrete, being a flow of electrons [43]. In the semiclassical picture, the probability of promoting an electron in the conduction band in a infinitesimal time interval is proportional to the light intensity. As a consequence, a plane wave with constant intensity in macroscopic time interval generates n_e photo-electrons following a Poissonian probability distribution P(n_e).

Indeed, in quantum mechanics a plane wave with constant intensity is represented by a coherent state of the form:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle |\beta\rangle = {\rm e}^{|\beta|^2/2}\sum_{n=0}^\infty\frac{\beta^n}{\sqrt{n!}}|n\rangle, \nonumber \end{align} \tag{ 1 }$$

with Poissonian photon number statistics $P(n)= |\langle \hat{n}|\beta \rangle|^2=$ ${\rm e}^{-\langle\hat{n}\rangle^2}\frac{\langle \hat{n}\rangle^n}{n!}$, which reproduces the observed photocurrent by the absorption of a photon and the excitation of a photo-electron. All the states that can be represented as a statistical mixture of coherent states, in Glauber–Sudarshan form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{G-S} \rho=\int {\rm d}^2 \beta P(\beta)|\beta\rangle\langle\beta|, ~\mathrm{with}~ P(\beta)\geqslant0, \nonumber \end{align} \tag{ 2 }$$

are 'classical' according to the definition conventionally used in quantum optics [44]. It is easy to show that these states must have Poissonian (or super-Poissonian) photon number fluctuation, with photon-number variance $\langle\Delta^2\hat{n}\rangle$ equal (or larger) to the mean photon number, i.e. $\langle\Delta^2\hat{n}\rangle \geqslant\langle\hat{n}\rangle$.

By contrast, the detection of sub-Poissonian variance, $\langle\Delta^2\hat{n}\rangle < \langle\hat{n}\rangle$, is sufficient to indicate the non-classical nature of the light, according to the quantum optics definition.

In order to study the photon statistics influence on the optical measurement, let us consider the problem of absorption estimation. Sending a probe beam with mean number of photon $\langle\hat{n}\rangle$ through a sample with absorption factor $\alpha$, and measuring the beam mean power after the interaction one gets $\langle\hat{n'}\rangle=(1-\alpha) \langle\hat{n}\rangle$, see figure 1(a). The uncertainty on the absorption it is simply given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{unc_prop} \Delta \alpha=\frac{\sqrt{\langle\Delta^{2} \hat n' \rangle}}{\left|\frac{\partial \langle n'\rangle}{\partial\alpha}\right|}. \nonumber \end{align} \tag{ 3 }$$

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If the probe statistics is Poissonian, immediately one obtains $\Delta \alpha_{\rm SNL}=[(1-\alpha)/ \langle\hat{n}\rangle]^{1/2}$, graphically represented by the red line in figure 1(b). Thus, it is clear how the SNL is related to the Poissonian fluctuation in classical states of light.

The phase estimation problem can be treated similarly. The output power of a Mach Zehnder or a Michelson interferometer is $\langle\hat{n'}\rangle=\tau \langle\hat{n}\rangle$, where $\tau$ depends on the phase shift $\phi$ among the arms, $\tau=\sin^{2}(\phi/2)$. Using the uncertainty propagation on equation (3) where $\phi$ is placed instead of $\alpha$, and assuming Poissonian statistics of the light, one gets $\Delta \phi= \langle\hat{n}\rangle^{-1/2} \cos(\phi/2){}^{-1}$. For $\phi=0$ it reaches the lowest value, which is again the SNL.

Such classical limits in loss and phase estimations that we have found here by assuming a specific and simple detection strategy and Poissonian light, coincide with the ultimate bounds achievable by a coherent state and the most general measurement allowed by quantum mechanics: see [42, 45, 46] for loss estimation and [39, 40, 47] for phase estimation.

Of course, increasing arbitrarily the number of photons, i.e. the optical power, one can always reduce the shot-noise contribution to the uncertainty budget below other technical noise. However, this is not always an option. Beating such a limit is particularly important when there are optical power constraints. In addition to fundamental problems related to quantum back-action [47], there are several practical situations in which one may need to limit the optical power.

• Extremely precise measurements. In the last generation of a gravitational wave detector the power circulating in the interferometer is of the order of 1 kW (CW), about 10²¹ photon s⁻¹. This allows reaching the sensitivity in the strain of $\Delta h<10^{-22} ~ \rm Hz^{-1/2}$. A further increasing of the power would have thermal effects on the surface of the mirrors and other optical components, producing an unwanted large scattering interfering with beam propagation.
• Probing delicate systems (biological sample, photosensitive chemicals). Damage due to optical tweezers has been reported for Escherichia coli, Listeria and other bacteria. Alteration of chemical and biological photosensitive processes can occur at very low power.
• Investigate response of a system at few photons level. Some biophysical and biochemical processes, for example phototransduction in vision, or photosynthesis are triggered by the absorption of one or few photons. Moreover, the development of quantum technologies are strongly based on single- or few-photon state manipulation and detection. In order to study and calibrate a single-photon detector, or the retina response at the few-photons level, it is fundamental to develop sub-shot-noise techniques.
• Developing standards and traceable measurements at the few-photons/single-photon level. The need for dedicated metrology tools in the few-photon regime is one of the keys to success of quantum photonics technologies (quantum cryptography, quantum sensing and imaging). On the other hand, new insights into the study of the retinal process and response at low light levels are a strong motivation for the development of quantum photometry.

In practice, realising a single mode beam with significant sub-shot-noise fluctuation reduction is quite challenging. Another way, more effective and commonly used in experimental demonstrations, relies upon the non-classical photon number correlation among two beams, the 'twin-beam' (TWB) state. The idea is that one beam of the pair is used as a probe while the other acts as a reference for the quantum noise, a property that can be exploited from interferometry [73, 74] to imaging [11, 12, 75].

SPDC is one of the most efficient ways to produce quantum correlations between optical fields. This physical phenomenon was discovered at the end of the sixties [76, 77] and occurs due to the interaction between an intense optical field, usually called a pump beam, and a non-linear dielectric optical medium [78]. Basically, the phenomenon consists in the decay of one photon of the pump into two photons, named 'signal' and 'idler' for historical reasons, preserving energy and momentum (see figure 2):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega_p &= \omega_1 + \omega_2 \nonumber \\ {\bf k}_p &= {\bf k}_1 + {\bf k}_2 \label{phasematch} \nonumber \end{align} \tag{ 6 }$$

where $\omega_p$ is the frequency of the pump photon and $\omega_1$ and $\omega_2$ are the frequencies of the signal and idler photons, respectively, and ${\bf k}_j$ (with $j=p,1,2$) are the corresponding wave vectors. Even though the emission of a pair is a quantum random process, not different from photon emission by a thermal source, the presence of one photon with a certain direction and frequency is bound to the presence of a 'twin' photon in a correlated spatial-frequency mode. In the high gain regime of SPDC, many photon pairs can be also generated occupying a plethora of bipartite correlated modes [12, 78]. Approximately this corresponds to the parallel generation of many entangled states, each formally written as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \vert {\rm TWB}\rangle_{1,2}=\sum_{n} c_{n}\vert n\rangle_{1}\vert n\rangle_{2}, \label{SPDC_state} \nonumber \end{align} \tag{ 7 }$$

where for simplicity the subscript '1' represents the spatial-frequency mode $({\bf q},\omega)$ and the subscript '2' represents the correlated mode $(-{\bf q},\omega_{p}- \omega)$ (${\bf q}$ is the transverse momentum) [78]. The probability amplitude is $c_{n} \propto \sqrt{\mu^n/(\mu + 1){}^{n+1}}$, where $\mu$ is the mean number of photons per mode.

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From the form of the coefficients c_n the super-Poissonian (thermal) character of the pair emission emerges. However, the thermal fluctuations are perfectly reproduced in the two modes. The degree of correlation is quantified by the noise reduction factor $\sigma$, as the ratio between the variance of the difference in the number of photons in the two modes, normalised to the corresponding shot noise [14, 79–88]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{NRF} \sigma=\frac{\langle\Delta^2(\hat{n_1}-\hat{n_2})\rangle}{\langle\hat{n_1}+\hat{n_2}\rangle}\equiv \frac{\langle\Delta^2\hat{n_1}\rangle+\langle\Delta^2\hat{n_2}\rangle-2 \langle\Delta\hat{n}_1\Delta\hat{n}_2\rangle}{\langle\hat{n_1}+\hat{n_2}\rangle}. \nonumber \end{align} \tag{ 8 }$$

For classical states, $\sigma$ is lower bounded by 1, while for a TWB in the form of equation (7) it is $\sigma_{\rm TWB} =0$. Also in this case the practical limit in the noise reduction is usually represented by the unavoidable presence of losses. From equation (4), considering two modes subject to the same transmission-detection efficiency $\newcommand{\e}{{\rm e}} \eta_1=\eta_2=\eta$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{NRF_eta} \sigma_{\rm det} = \eta \sigma+ 1-\eta. \nonumber \end{align} \tag{ 9 }$$

The lower bound in the presence of losses is therefore $\newcommand{\e}{{\rm e}} \sigma_{\rm det} = 1-\eta$.

The sub-shot-noise (SSN) correlation of the TWB state has been experimentally demonstrated both in the case of two-mode state as in equation (7) [82, 84, 89–91] and in the case of many spatial modes detected in parallel by the pixels of a CCD camera [79, 81, 92].

SSN correlations can also be efficiently generated by four-wave-mixing (FWM) in atomic vapours [93–96], a process depicted in figure 3. A double-$\Lambda$ configuration with three atomic levels is exploited (often the hyperfine splitting structure of rubidium), so that when two pump photons are absorbed, two photons, the signal and the conjugate, are generated ($\omega_{\rm conj}=\omega_{\rm sign}+2\delta$, where $\delta$ is the hyperfine energy splitting). Given the intrinsic high gain of the process, and the stimulation obtained by seeding the signal with a macroscopic beam, FWM leads to the production of intense quantum correlated beams that can reach noise reduction $\sigma <1$ for several tens of ${\rm \mu}$W of optical power.

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Based on this strong non-classical correlation, TWB states have shown the possibility of sub-SNL sensitivity in absorption/transmission measurements [87, 97–100], quantum ellipsometry [101, 102], quantum enhanced sensing [9, 10, 103, 104], quantum reading of digital memories [105] and plasmonic sensors [106, 107]. The common idea behind these works is that the random intensity noise in the probe beam addressed to the sample can be known by measuring the correlated (reference) beam and subtracted.

These considerations can be extended to the multi-mode case. Indeed, when TWBs are produced through parametric down conversion, or by FWM in atomic vapours [96, 108, 109], the emission is spatially broadband, forming a collection of pairwise correlated modes in the transverse plane. Modern high sensitivity multi-pixel detectors, like charge coupled devices (CCDs), complementary metal-oxide semiconductor (CMOS) cameras, or single-photon detector arrays, can detect simultaneously thousands of correlated spatial modes with high efficiency, improving the sensitivity of imaging applications even in wide-field modality [6, 110–112].

One of first application of SPDC entangled photons has been 'ghost imaging' (GI) [113, 114], whose goal is the reconstruction of the spatial transmission/reflection profile of an object even if the interacting photons are collected by a single pixel detector. After the first realisation, it has been shown that GI can also be obtained with classical beams, nevertheless non-classical correlations provide some advantages at very low photon flux [115–117].

Finally, non-classical correlations have disclosed new possibilities in quantum radiometry [118], e.g. the possibility of absolute calibration of detectors without the need for comparison with calibrated standards. The first proposal for calibrating single-photon detectors was formulated by Klyshko [76] just after the discovery of the SPDC process and nowadays it is an established technique [13] currently used in metrological institutes. Generalising the method to the domain of analogue detectors and spatially resolving detectors has recently led to the absolute calibration of electron multiplying CCD cameras (EMCCDs) from the linear regime [14, 119] to the on–off single-photon regime [15], by just changing the intensity of the SPDC pump laser, as well as of ICCD cameras [120, 121].

Single-photon sources, and more generally Fock state sources, are necessary for applications in the field of quantum technologies (e.g. quantum computing, quantum key distribution and quantum metrology), which are among the most relevant topics with respect to innovation and high technology worldwide. An ideal single-photon source emits one photon on demand, at a time chosen by the user, with the emitted photons being indistinguishable from one another and having an adjustable repetition rate [122–124].

Based on these requirements, two figures of merit are generally used for characterising the quality of a single-photon source. They are represented in figure 4. The Hanbury-Brown and Twiss (HBT) interferometer in figure 4(a) measures the tendency of the photons to arrive in pairs. Formally, the HBT experiment allows the evaluation of the second order Glauber correlation function, defined as $\newcommand{\e}{{\rm e}} g^{(2)}(\tau=0)\equiv \langle \hat n(\hat n-1)\rangle/\langle \hat n\rangle^{2}$. For coherent light (a stable laser) this parameter equals unity; for thermal light it turns out to be 2; while it must be zero if the photons are emitted one by one. Thus, for a single-photon source, the coincidence level between the two detectors at the varying of the temporal delay $\tau$ presents a dip exactly at $\tau=0$. The second scheme, sketched in figure 4(b) and known as the Hong–Ou–Mandel (HOM) interferometer, measures the indistinguishability of two photons produced at different times by a single-photon source. Only if the photons (or more precisely their paths) cannot be distinguished, even in principle, with respect to any degrees of freedom (polarisation, wavelength, spatial mode, etc), quantum interference happens, forcing the photons to exit always from the same side of the beam splitter and no coincidences are registered. If the two photons can be identified, for example temporally, the interference disappears. Therefore, only for $\tau=0$ a dip is observed.

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A Fock state source [43] is a generalisation of a single-photon source, since it should be able to emit a fixed number of photons on demand that are indistinguishable from one another. Obviously, a Fock state source emitting n photons per pulse can be realised by exploiting n ideal single-photon sources operating together. Such photon sources have the potentiality to become a new quantum standard with a huge range of applications: calibration/characterisation of single-photon counter devices [125–128], realisation of the SI base unit candela [118], quantum enhanced measurements [47], quantum sensing [103, 129] and quantum imaging.

In recent years, there has been significant progress in the field of single-photon sources and their use in metrology, also thanks to the research activities started in national metrological institutions aimed at developing metrological techniques for raising quantum technology applications exploiting single photons. In particular, single-photon sources were calibrated in a traceable way with respect to their absolute optical radiant flux and spectral power distribution [130, 131]. Although this might be considered as a huge step towards the realisation of a new photon standard source, the photon fluxes are still too low, and the purity is still insufficient for real practical use.

Indeed single-photon sources, typically, present intrinsic single-photon emission losses due to non-radiative decay mechanisms allowing the spontaneous decay of a single excited quantum emitter, without the emission of the expected single photon [23, 24]. These non-radiative mechanisms dissipate the energy associated with the quantum emitter decay in some other way, for instance in the form of phonons in solid state systems. In some cases, it is possible to effectively suppress such non-radiative decays (e.g. operating the emitter at cryogenic temperature), obtaining an almost ideal single-photon source showing nearly 100% quantum efficiency [22, 132, 133]. Even in the presence of such ideal single-photon sources, the photon flux emission is often limited by the poor optical collection and out-coupling of the emitted single photons.

To enhance the photon fluxes of single-photon sources, and make them closer to a deterministic behaviour, photonic structures for enhancing the photon collection efficiencies are required. Indeed, even exploiting high numerical aperture objectives, the collection efficiency is of the order of a few per cent. Several photonic structures for photon collection enhancement are under investigation, between them solid immersion lenses, waveguiding structures, micro-resonators or planar optical antennas [134–144].

The actual single-photon sources are still far from being completely predictable, deterministic and indistinguishable. Improvements are required to bring single-photon sources close to the ideality, but they can be already exploited in several interesting applications in the field of quantum technologies in general, and for quantum imaging applications in particular, because of their intrinsic anti-bunching or sub-shot noise properties [145].

By contrast, non-radiative and optical losses as well as decoherence limit the applicability of non-ideal sources to scenarios where deterministic emissions of ideal and completely identical single photons are necessary, such as, e.g. when interference/interaction between two (or more) single photons is exploited, as in quantum computation or simulation. Moreover, from the metrological point of view, ideal single-photon sources on demand are highly desirable for developing absolute sources with finite photon number emission without intensity fluctuation. This could lead to a redefinition of radiometric and photometric units in terms of the number of photons (quantum Candela) [118].

We highlight that a reasonable approximation of single-photon sources are the heralded photons produced by SPDC [146, 147]. As mentioned in section 3.1, photons are always emitted in pairs with low probability, but one can get rid of the vacuum component since the detection of one photon of the pair heralds the presence of the other one, and the probability of double pair emission remains very low. These pseudo single-photon sources, based on SPDC or FWM, have interesting performances in terms of photon rate [124] and quality of the single-photon emission, but they typically need temporal post-selection [124] to distinguish the heralded photon from a dominant unheralded background photon (even if there are remarkable exceptions [146, 148]).

Despite the fact that temporal post-selection is a limiting factor for the possibilities of practical exploitation of heralded single-photon sources in quantum technologies, this approach has been demonstrated recently for quantum enhanced absorption measurement and spectroscopy of a biological sample (haemoglobin) with post-selection of the heralded single photons [149], and with selection performed by active feed-forward enabled by an optical shutter [150].

When it is possible to exploit temporal post-selection, e.g. for pulsed TWB emission, we should note that TWB, multi-modes or in the high gain regime, can be exploited as an heralded Fock state source, having on the heralding arm a photon number resolving detector with (nearly) ideal unit efficiency.

Fluorophores and markers used in biological microscopy can be chosen among single-photon emitters such as quantum dots, dye molecules or NV centres in (nano)diamond. A single-photon emitter presents, by definition, a strong anti-correlation in the temporal photon emission, since at any time the presence of more than one photon is prevented by the excitation-decay process (see figure 4 and its discussion in the text).

This effect is known as anti-bunching and can be exploited in many applications, from quantum information and communication to metrology and super-resolution imaging. A possible approach exploits the measurement of the higher order Glauber correlation function at $\tau=0$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g^{(k)}(\tau=0)=\frac{\langle\prod\nolimits_{i=0}^{k-1}(\hat{n}-i)\rangle}{\langle \hat{n} \rangle^k}, \nonumber \end{align} \tag{ 10 }$$

in each position of the image plane, obtaining an increase in the resolution of the single-photon emitters map. Indeed if two or more single-photon emitters are closer than the DL, thus indistinguishable from the standard fluorescence intensity map, the presence of coincident single-photon detection indicates the presence of more than one single-photon emitter. This additional information, together with the intensity map, allows the reconstruction of a super-resolved map [151–153]. In general, for an arbitrary number of centres in the cluster, a resolution improvement factor of $1/\sqrt{k}$ is achieved when the Glauber autocorrelation function is measured up to the kth order (g^(k)).

Experimental demonstrations of this technique have been performed in wide-field and in confocal microscopy [152, 153]. Figure 5 presents some of the results obtained in [153], by exploiting NV centres in a single-photon confocal microscope.

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Recently it has been proposed to combine this high order correlation function method with the use of structured excitation light [154]. It has been shown that in this case one could reach a resolution increase of $k+ \sqrt{k}$ with respect to the DL.

Another technique, realised in 2017 [155], exploits anti-bunching together with the natural 'photoblinking' effect found in many types of fluorophores and a fibre bundle collecting light with certain spatial resolution. After the reconstruction, a final resolution of 20 nm was reported.

It is worth mentioning the research on sub-diffraction imaging triggered by the interesting paper by Tsang and coworkers [156]. In their quantum-estimation-inspired proposal [157], they observe that, given only two single-photon emitters, a lot of information is present in the sum of the electromagnetic field of the two emitters at the image plane rather than just the sum of the intensity alone. Thanks to this, and using the quantum estimation theory, they were able to devise imaging techniques able to beat by far the diffraction limit. The idea of identifying the ultimate bound of the ability to estimate the distance between a pair of closely separated sources, achieving near-quantum-limited performance has inspired a relevant amount of theoretical researches (see e.g. [158–161]), and also a certain amount of proof-of-principle experiments (see e.g. [162–166]). It is important to notice that all these results are limited to the case of only two sources; it is far from being obvious that similar performance can be achieved in the case of three or more sources. It is expected that this becomes an active research field in the future years also because of its inherent connection with the multi-parameter quantum estimation problem [157, 167].

Extracting relevant information about a delicate sample with the lowest photon dose is of paramount importance in biochemistry and biology to ensure that the processes being investigated are not shifted to an alternate pathway due to environmental stress. Wide field microscopy is the simplest, fastest, least expensive and oldest imaging solution used, for example, for live-cell imaging and it has the advantage of requiring the lowest photon dose, especially in transmission imaging with respect to scanning confocal microscope [168].

The exploitation of multi-mode correlations in a TWB has been proposed for high sensitivity wide field imaging of a weak absorbing object in [110] and a proof of principle of this technique has been reported by Brida et al in 2010 [111]. Recently, the first wide-field sub-SNL microscope [6] was realised, providing images of 10⁴ pixels with a true (without post-selection) significant quantum enhancement, and a spatial resolution of few micrometres. This represents a considerable advancement towards a real application of quantum imaging. In the following we will discuss some details of this application.

We have already seen that sub-Poissonian light can be used to achieve SSN absorption estimation according to equation (5). However it is challenging to experimentally produce single modes with sub-Poissonian photon statistics. A different approach is to consider two correlated modes from a TWB state, and using one of them as a reference for the noise in a 'differential' scheme, as presented in figure 6(a). In this case the uncertainty is:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Ualpha} \Delta \alpha_{\rm TWB}\simeq \sqrt{\frac{\alpha(1-\alpha)+ 2\sigma_{\rm det}(1-\alpha)^{2}}{\langle N_{\rm P} \rangle}}, \nonumber \end{align} \tag{ 11 }$$

where $\sigma_{\rm det}$ is the the noise reduction parameter in equation (9). For a TWB it equals the losses in detecting correlated photons, $\newcommand{\e}{{\rm e}} \sigma_{\rm det}=1-\eta$. Ideal lossless detection leads to the ultimate quantum limit, $\newcommand{\e}{{\rm e}} \Delta \alpha_{\rm TWB} = \sqrt{\alpha}\Delta \alpha_{\rm SNL}\equiv \Delta \alpha_{\rm UQL}$, which is represented by the black curve in figure 1(c). The performance of a TWB in loss estimation has been theoretically discussed in [97–100] and demonstrated experimentally in [87, 99, 100]. In [100] more than 50% sensitivity improvement with respect to the SNL was reported, and it reached a 100% improvement when compared with conventional two-beam approach, represented in figure 6(b). Note that the two-beam approach is extensively used in standard devices like spectrophotometers, where a classical beam is split in two and one beam is used to monitor the instability of the source and detectors and to compensate for them. However, in classical correlated beams (CCBs) generated in this way, only the super-Poissonian component of the fluctuations is correlated (sometimes called classical 'excess noise'), whereas the shot noise remains uncorrelated and cannot be compensated. Indeed the TWB represents the natural extension of this method to the quantum regime. Recently similar schemes, but exploiting four wave mixing in rubidium vapour, have been applied for monitoring index refraction change in plasmonic sensors [106, 107].

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The extension of the technique from the two-mode case to the multi-mode case allows the realisation of wide field SSN imaging, as represented schematically in figure 7. Basically, it is possible to match the pixel structure of a spatially resolving detector with the spatial distribution of the multi-mode TWB generated in the SPDC process, so that each pair of correlated spatial modes is precisely and entirely detected by a pair of pixels. The quantum correlation in transverse momentum, between ${\bf q}$ and ${\bf -q}$, is converted in spatial correlation among symmetric pixels in position ${\bf x}$ and ${\bf -x}$ at the object plane by a far-field lens (not shown in the figure). The noise of the image taken in one branch is removed pixel-by-pixel by 'subtracting' the noise pattern measured on the other branch [111]. In [6], the method has been demonstrated in a microscope configuration, using a CCD camera operating in linear regime, with high quantum efficiency (95%) and low noise (few electrons per pixel per frame). The experimental evidence of the noise reduction and the sensitivity improvement obtained is presented in figure 8.

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The spatial resolution of this technique is given by the traverse size of the correlation (coherence area), namely the uncertainty in the relative propagation direction of correlated photons, which is proportional to the inverse of the pump width. Pixel size should by large enough to include at least a coherence area, otherwise correlated photons may be lost affecting the noise reduction factor. In fact, figure 8 shows that at full resolution (d  =  1) the noise reduction is modest while it becomes better if the resolution decreases.

Finally, we mention that several proofs of principle- of quantum enhanced phase-contrast scanning microscopy exploiting NOON states (N  =  2) [62–65] have been reported, although, in most of the cases, a significant enhancement without post-selection or losses compensation is still missing.

Sub-shot-noise photon number/intensity correlations enable detection of the displacement of a light beam with accuracy surpassing standard methods based on laser beams. The standard way to monitor the wandering of a light beam is to send it to a quadrant detector as depicted in figure 9(a). This configuration is used for measuring small displacements in many applications, for example, in atomic force microscopy, ultra-weak absorption measurements, or single molecule tracking in biology [5, 71]. When the beam is perfectly aligned to be symmetric with respect to the origin of the axis, the difference in the power measured in the different sectors is, on average, null. Small displacement in one direction, for instance along the x-axis, will give a proportional change in the the photocurrent difference [169],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{beam_disp} \langle n_{A+C}-n_{B+D}\rangle= \frac{2}{\pi^{1/2}} \frac{\delta x}{w}N, \nonumber \end{align} \tag{ 12 }$$

where w is the waist of the beam (let us suppose a gaussian beam) and $N=n_{A+C}+n_{B+D}$ is the total photon number in the beam. However, due to the shot noise, which is independent between right and left sectors, the noise on the right side of equation (12) is exactly N^1/2 and the minimum detectable displacement becomes:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{D_x_SNL} \Delta x= \left(\frac{\pi^{1/2} w}{2}\right) \frac{1}{\sqrt{N}}. \nonumber \end{align} \tag{ 13 }$$

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It is interesting to note that the spatial resolution limit in this case is given by the energy available in the beam, or the power limit supported by the quadrant detector.

Multi-mode quantum correlations allow beating the SNL in equation (13). Following previous theoretical investigation [170], the first demonstration of SSN displacement detection along one dimension was obtained in [171] by combining vacuum squeezed beam and a coherent beam that were spatially orthogonal. Although the resultant beam is not squeezed globally, it is shown to have strong (non-classical) internal spatial correlations between the portions of the beam impinging different sectors of the detector. The technique has been extended in the 2D case in [72].

An evolution of this technique has led to the experimental demonstration of sub-shot-noise particle tracking in living systems and microrheology within Saccharomyces cerevisiae yeast cells [71]. Lipid granules were tracked in real time as they diffused through the cytoplasm surpassing the SNL of 42%. In typical laser-based particle tracking, the presence of a particle causes light to be scattered out of an incident field. The interference between scattered and transmitted fields leads to a deflection of the incident field proportional to the displacement of the particle. The difficulties of using quantum correlation in this context are that, on one side such measurements are typically conducted at low frequencies where technical noise is dominant with respect to the shot noise, and on the other side, the distortion of the spatial mode propagating through high-numerical optical system and biological samples prevents the quantum light from matching the detection mode (quadrant detector). To circumvent these issues, in [71] two separate fields are used, one for interrogation and the other a 'flipped' Gaussian local oscillator to define the 'detection mode'. Interference among them, measured by a single photodiode, provides particle information equivalent to a quadrant photodiode.

In 2015, Pooser et al demonstrated a noise reduction of 60% below the SNL in state-of-the-art displacement sensitivity for a MEMS cantilever [9]. They used a quantum beam of power up to hundreds of ${\rm \mu}$W reaching a few fm ${\rm Hz}^{-1/2}$ resolution in the region of 10⁵ kHz. Figure 9(b) presents the sketch of the experiment where two spatially multi-mode correlated TWBs, generated by FWM, are superimposed at a split detector. As we discussed in section 4.3, the variance of the photocurrent difference between symmetric portions of the two beams in figure 9(c), either A and B' or A' and B, is lower than the SNL. Moreover, this is true also when they are spatially overlapped at the same detector. Therefore the fluctuations of the left side of equation (12) are reduced below the SNL and this is the origin of the advantage of the technique.

We mention also that a similar scheme has been investigated in the discrete case, considering spatially entangled biphoton pairs [172]. In principle, it has been found that the smallest resolvable parameter of a simple split detector scales as the inverse of the number of pairs (Heisenberg scaling) when this number is very small.

In a different perspective, but conceptually analogous to the previous case, there is the estimation of the displacement of a shadow casts by a fully opaque object intercepting a probe beam. In [173], a noise reduction corresponding to an improvement in position sensitivity of up to 17% was obtained with bi-photon pairs, created with SPDC and detected by an EMCCD camera employed as a photon-number-resolving split detector. As reported in the literature [15], the EMCCD can be operated as a photon number resolving detector, thanks to spatial multiplexing, where each pixel is operated in the 'on–off' regime by setting a discriminating threshold.

Quantum ghost imaging (QGI) and ghost-spectroscopy (QGS) are two complementary techniques based on signal and idler photons correlations in SPDC (or FWM).

QGI was proposed in 1994 [113] and experimentally realised in 1995 by Pittman et al [114]. It makes the reconstruction of the transmittance (or reflectance) profile of an object, placed in the signal beam, possible, although the light after the interaction is collected by a single pixel or 'bucket' detector, i.e. without spatial resolution. The spatial information is retrieved by performing correlation measurements (e.g. temporal coincidence) among the single pixel detector and each pixel (or position) of a spatially resolving (or scanning) detector on the idler beam. The working principle is the following: the spatial selection performed in the idler beam automatically identifies a small range of allowed positions for the correlated signal photons at the object plane, expressed by a function $\Gamma(x_{i}+ x_{s})$ peaked into zero and which has to be narrower than the minimum size of the object variation scale. If a coincidence is registered between the two detectors, this means that in the spatial resolution cell identified in that way, the object is transmitting. On the other hand, if the coincidence is missed, the object has absorbed the signal photon. In this way, by measuring the number of coincidences R(x_i) in function of the position x_i of the idler pixel, it is possible to reconstruct the whole transmission T_s of the object in the signal path, $R(x_{i})\propto\int_{\rm Bucket} \Gamma(x_{i}+ x_{s}) T_{s}(x_{s}) {\rm d}x_{s} \simeq T_{s}(-x_{i})$. Instead of measuring time coincidences between SPADs operated in on–off modality, it is also possible to measure temporal correlation of the intensity fluctuations between a couple of linear detectors if they are sensitive enough to register such signals, which are usually faint.

A lively debate as to whether ghost imaging truly requires quantum light started just after its first demonstration. Many theoretical investigations [174–176] and experiments [177–181] have demonstrated that almost all the quantum ghost imaging features can be mimicked by classical correlation, usually generated by splitting a spatially incoherent single pseudo-thermal beam (see figure 10) obtained from a laser beam scattered by a rotating ground-glass disk (not shown in figure). However, thermal GI can be obtained with sunlight [182]) or a broadband super-luminescent diode source [183] although the short coherence time make the detection scheme rather difficult. An interesting compromise between thermal-light-based and TWB-based GI can be found in [184], where the authors use speckle correlation in an intense TWB seeded with thermal light.

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Even computational techniques allow GI reconstruction [185–187], by impressing random spatial patterns on a single beam modulated in a controllable and known way (spatial light modulators, either in phase or amplitude are standard devices currently used in many imaging applications).

Since the first proof-of-principle demonstrations, either with TWBs or with classical light, substantial advancements towards the applicability of GI have been achieved: drastically improving the SNR by differential GI [188], measuring reflected photons [189], using polychromatic light [190, 191], and extending GI to 3D reconstruction [192]. For enhancing visibility and contrast of GI, the use of high-order correlations has been proposed and applied [180, 193, 194]. Also compressive sensing techniques have been successfully transferred to GI, speeding up the image reconstruction [187, 195].

There is a really huge amount of literature on GI, both concerning its relation with more fundamental questions about the quantum-classical boundary and applications. We do not intend here to be exhaustive and we advise the interested reader to refer to dedicated reviews [176, 196].

Among the general advantages of ghost imaging that have intrigued scientists there is robustness against effects like scattering and phase distortion that are experienced between the object and the bucket detector. This can have important applications in an open-air ranging system or for inspecting biological samples, where tissues represent the diffusive medium. In fact, even if the propagation directions of photons are scrambled after the sample, it does not make any difference to the bucket detector, which integrated in any case. Many works have demonstrated turbulence free ghost imaging and ghost imaging through turbid media [188, 197–199].

Another possibility offered by QGI that is difficult to emulate with classical sources is 'two-colour' GI [200–202]. By producing pairs of non-degenerate photons, the object can be illuminated by photons with a significantly different wavelength from the ones detected with spatial resolution on the second beam. For example, image reconstruction in the infra-red range can be achieved with cameras operated in the visible spectrum [203]. Moreover, GI is always an option in all the situations in which space constraints or 'hostile' environment conditions, such as e.g. extreme temperature or high electromagnetic fields, do not allow to set-up an imaging system with a spatially resolving detector close to the sample. As proof of principle, in [204] the authors apply the GI approach to image the magnetic domain structure of an yttrium iron garnet sample by exploiting the Faraday effect, opening up possible development of magneto-optical imaging at extreme cryogenic temperatures or in the presence of high magnetic fields.

In conclusion, one of the big advantages in using quantum correlation is given by the ability of rejecting external technical noise and, in general, uncorrelated background, especially when faint light levels are required [7, 116, 117]. Furthermore, note that the noise can also be due to uncorrelated photons produced by the same source used for GI. Indeed, all the spatio-temporal modes collected by the bucket detector that are not correlated with the spatial-mode impinging on a specific pixel of the spatially resolving detector contribute to the uncorrelated background.

Basically, since the photons are produced in pairs at the same time the coincidences among two SPADs appear within a very short time window $\Delta t$ (typically a few ns, because of the detectors and correlation/coincidence-electronic jitter). The number of 'true' detection events seen by each detector is simply given by $\newcommand{\e}{{\rm e}} N^{(T)}_{i(s)}=\eta_{i(s)}P \Delta t$, where P is the pairs production rate. Analogously, the background counts are $N^{(B)}_{i(s)}=B_{i(s)}\Delta t$ with B_i(s) the rate of background counts. The number of true coincidences is proportional to the probability of joint detection by the two single-photon detectors (located in the two channels) $\newcommand{\e}{{\rm e}} R^{(T)}=\eta_{i}\eta_{s} P \Delta t$. Assuming that noise photocounts of the two single-photon detectors are uncorrelated, the number of accidental coincidences R^(A) is given by the product of the probability components due to accidental overlap of SPDC and noise photocounts, $\newcommand{\e}{{\rm e}} R^{(A)}=N^{(B)}_{i} N^{(B)}_{s}+N^{(B)}_{i} N^{(T)}_{s}\,+\,N^{(T)}_{i}N^{(B)}_{s}=[B_{i}B_{s}+(\eta_{i}B_{i}\,+$$\eta_{s}B_{s})P](\Delta t){}^{2}$, where we observe that it scales as the square of the detection time window. Let us suppose that the background rate is dominant, i.e. $N^{(B)}_{i(s)}\gg N^{(T)}_{i(s)}$. The signal to noise ratio of quantities proportional to the true coincidence is $\newcommand{\e}{{\rm e}} {\rm SNR}_{\rm QGI}=R^{(T)}/R^{(A)}\simeq\eta_{i}\eta_{s}P/(B_{i}B_{s}\Delta t)$. This performance should be compared with the direct single branch case in which the SNR is $\newcommand{\e}{{\rm e}} {\rm SNR}_{D}=N^{(T)}_{s}/N^{(B)}_{s}\simeq\eta_{s}P/B_{s}$. Therefore, the advantage in terms of SNR of the quantum correlation technique is $\newcommand{\e}{{\rm e}} {\rm SNR}_{Q}/{\rm SNR}_{D}=\eta_{i}/(B_{s}\Delta t)$ [205]. This means that QGI delivers better image quality reconstruction when the efficiency of the reference (idler) arm is high and the number of photons in the detection time is smaller than unity. This also remains valid in the case of analogue detection, in which, rather than temporal coincidence, intensity correlations are measured, as derived theoretically and demonstrated experimentally in [117]. In [7], QGI of a wasp wing was obtained by less than 0.5 photon detected per pixel, with the help of a dedicated compressive sensing algorithm. In that case an intensified CCD camera was gated by the photons arriving at a SPAD, which played the role of the single pixel detector. An imaging preserving delay line is necessary to compensate for the time necessary for the electronics to send the triggering input to the camera.

Similar to ghost imaging, in quantum QGS, the correlation in frequency shown in equation (6) allows the retrieval of the absorption spectrum of $T_{s}(\omega_{s})$ by measuring the temporal coincidences among photon pairs $R(\omega_{i})\propto T_{s}(\omega_{p}-\omega_{i})$, and the spectral selection can be done in the beam that does not interact with the object. The first proposals and experiments have been reported in [206] and [207]. This scheme provides the same advantages of ghost imaging, offering the possibility of spectral selection in range significantly different with respect to the one of the light impinging the sample [208]. The second benefit is that the spectral profile can be obtained by exploiting an extremely low signal level compared to external and technical noise [205]. Again, it is very promising in all those applications in which delicate or photosensitive systems require the lowest dose of photons, such as in biological spectroscopy. We mention that even ghost spectroscopy can be realised with classical light, but in a completely different regime [209].

4.5.1. Imaging and spectroscopy without photon detection.

4.5.2. Quantum illumination.