Photon-number correlation for quantum enhanced imaging and sensing

By analogy with equation (7), a classical two-mode (bipartite) state is represented by a Glauber–Sudarshan probability density function $P({\alpha }_{1},{\alpha }_{2})\geqslant 0$

$$\begin{eqnarray}&&{\rho }_{\mathrm{1,2}}=\int {{\rm{d}}}^{2}{\alpha }_{1}{{\rm{d}}}^{2}{\alpha }_{2}P({\alpha }_{1},{\alpha }_{2})| {\alpha }_{1}\rangle | {\alpha }_{2}\rangle \langle {\alpha }_{1}| \langle {\alpha }_{2}| .\end{eqnarray} \tag{ 11 }$$

We can quantify the degree of correlation between the modes and their non-classical features defining the noise reduction factor (NRF) σ as the ratio between the variance of the difference in the number of photons and the noise of two coherent states of equivalent energies [81–91]:

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{\langle {{\rm{\Delta }}}^{2}({\hat{n}}_{1}-{\hat{n}}_{2})\rangle }{\langle {\hat{n}}_{1}+{\hat{n}}_{2}\rangle }=\displaystyle \frac{\langle {{\rm{\Delta }}}^{2}{\hat{n}}_{1}\rangle +\langle {{\rm{\Delta }}}^{2}{\hat{n}}_{2}\rangle -2\langle {\rm{\Delta }}{\hat{n}}_{1}{\rm{\Delta }}{\hat{n}}_{2}\rangle }{\langle {\hat{n}}_{1}+{\hat{n}}_{2}\rangle }.\end{eqnarray} \tag{ 12 }$$

The NRF represents the equivalent of the Fano factor for a bipartite state; in this case, the shot-noise level is given by the sum of the shot noise of the two modes $\langle {\hat{n}}_{1}+{\hat{n}}_{2}\rangle$. For classical bipartite states, σ is larger than or equal to 1. For non-classical beams, quantum correlations can lead to $0\leqslant \sigma \lt 1$. As for the Fano factor, σ is affected by the optical losses experienced by the two fields. From equation (3), considering two modes subject to the same transmission-detection efficiency ${\eta }_{1}={\eta }_{2}=\eta$,

$$\begin{eqnarray}&&{\sigma }_{{\rm{\det }}}=\eta \sigma +1-\eta .\end{eqnarray} \tag{ 13 }$$

The lowest bound in the presence of losses is therefore ${\sigma }_{{\rm{\det }}}=1-\eta$.

A demonstration of the classical limit of the correlation is given here in a specific case. Let us consider a two-mode state generated by splitting a single mode $\hat{a}$ with a BS of transmittance τ. In the case of ideal photodetection, the statistics of the output modes can be computed using the input–output relations of the BS in equation (1) (with $\tau =\eta$) as:

$$\begin{eqnarray}\langle {{\rm{\Delta }}}^{2}{\hat{n}}_{1}\rangle & = & \langle {\hat{{b}_{1}}}^{\dagger }\hat{{b}_{1}}{\hat{{b}_{1}}}^{\dagger }\hat{{b}_{1}}\rangle -\langle {\hat{{b}_{1}}}^{\dagger }\hat{{b}_{1}}{\rangle }^{2}\\ & = & {\tau }^{2}\langle {{\rm{\Delta }}}^{2}\hat{n}\rangle +\tau (1-\tau )\langle \hat{n}\rangle \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}\langle {{\rm{\Delta }}}^{2}{\hat{n}}_{2}\rangle & = & \langle {\hat{{b}_{2}}}^{\dagger }\hat{{b}_{2}}{\hat{{b}_{2}}}^{\dagger }\hat{{b}_{2}}\rangle -\langle {\hat{{b}_{2}}}^{\dagger }\hat{{b}_{2}}{\rangle }^{2}\\ & = & {(1-\tau )}^{2}\langle {{\rm{\Delta }}}^{2}\hat{n}\rangle +\tau (1-\tau )\langle \hat{n}\rangle \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\langle {\rm{\Delta }}{\hat{n}}_{1}{\rm{\Delta }}{\hat{n}}_{2}\rangle & = & \langle {\hat{{b}_{1}}}^{\dagger }\hat{{b}_{1}}{\hat{{b}_{2}}}^{\dagger }\hat{{b}_{2}}\rangle -\langle {\hat{{b}_{1}}}^{\dagger }\hat{{b}_{1}}\rangle \langle {\hat{{b}_{2}}}^{\dagger }\hat{{b}_{2}}\rangle \\ & = & \tau (1-\tau )[\langle {{\rm{\Delta }}}^{2}\hat{n}\rangle -\langle \hat{n}\rangle ].\end{eqnarray} \tag{ 16 }$$

The last expression reveals that in order to have a non-null covariance the statistics of the incoming light must be super-poissonian, which also means that a split coherent state does not generate any correlation, while a thermal beam does. Using the relations (14)–(16) in equation (12) one can express the NRF as:

$$\begin{eqnarray}&&\sigma =(F-1){(2\tau -1)}^{2}+1\end{eqnarray} \tag{ 17 }$$

where $F=\langle {{\rm{\Delta }}}^{2}\hat{n}\rangle /\langle \hat{n}\rangle$. For a balanced 50:50 ($\tau =1/2$) BS this leads to the classical limit $\sigma =1$, irrespective of the statistical properties of the incoming beam, either sub-poissonian or super-poissonian. This means that the fluctuations of the two modes are suppressed by the subtraction, except the shot noise. On the other hand, for unbalanced BS, i.e. $\tau \ne 1/2$, an incoming field with sub-poissonian statistics generates non-classical correlations ($\sigma \lt 1$) at the output ports.

Another parameter that can be used as an indicator of non-classicality for two-mode states is the Cauchy–Schwarz parameter [92]:

$$\begin{eqnarray}&&\varepsilon =\displaystyle \frac{\langle :{\rm{\Delta }}{\hat{n}}_{1}{\rm{\Delta }}{\hat{n}}_{2}:\rangle }{\sqrt{\langle :{\rm{\Delta }}{\hat{n}}_{1}:\rangle \langle :{\rm{\Delta }}{\hat{n}}_{2}:\rangle }}\end{eqnarray} \tag{ 18 }$$

where $\langle ::\rangle$ is the normally ordered quantum expectation value. While σ is deteriorated by the losses, ε is remarkably immune to them, and for this reason it allows experimental access to the non-classical features, even for inefficient detection processes. However, noise added to the detection degrades its value (see section 6). For classical states of light, with a positive Glauber–Sudarshan P function, the Cauchy–Schwarz parameter is $\epsilon \leqslant 1$, while for states with a negative (or singular) P function this limit can be violated. In the case of correlated thermal beams, obtained by a 50:50 BS, the most used classically correlated states (for example, in the classical ghost-imaging protocols, see section 7), $\langle :{{\rm{\Delta }}}^{2}{\hat{n}}_{1}:{\rangle }_{{\rm{TH}}}\,=\langle :{{\rm{\Delta }}}^{2}{\hat{n}}_{2}:{\rangle }_{{\rm{TH}}}=\langle {\rm{\Delta }}{\hat{n}}_{1}{\rm{\Delta }}{\hat{n}}_{2}{\rangle }_{{\rm{TH}}}=\langle \hat{n}{\rangle }^{2}$, as can be simply derived by equations (14) and (16), by introducing the thermal variance $\langle {{\rm{\Delta }}}^{2}\hat{n}\rangle =\langle \hat{n}\rangle (1+\langle \hat{n}\rangle )$. Therefore, the Cauchy–Schwarz parameter for a split thermal beam is ${\varepsilon }_{{\rm{TH}}}=1,$ saturating the classical bounds. This demonstrates that thermal split beams show the best possible correlation allowed for classical states. They represent the classical benchmark for comparing the quantum enhanced performance in some emblematic imaging and sensing protocols, see sections 6 and 7.

In non-linear optics, the dielectric polarization P is expanded as [69, 123]:

$$\begin{eqnarray}&&P={\chi }^{1}E+{\chi }^{2}{EE}+{\chi }^{3}{EEE}+....\end{eqnarray} \tag{ 20 }$$

For higher strengths of the electric field (E), the higher-order non-linear terms become important. In addition to ${\chi }^{1}$ being the linear susceptibility coefficient, ${\chi }^{2}$, ${\chi }^{3}$ (...${\chi }^{n}$) are called the non-linear susceptibility coefficients of the medium. Taking into accounts non-linear effects up to second order, the field Hamiltonian in a non-magnetic medium, $H={\int }_{V}\tfrac{1}{2}\vec{E}\cdot ({\epsilon }_{0}\vec{E}+\vec{P})$ can be written as:

$$\begin{eqnarray}&&H(t)={\int }_{V}\left[\displaystyle \frac{1}{2}{\epsilon }_{0}{E}^{2}(r,t)+{X}_{1}(r,t)+{X}_{2}(r,t)\right]{\rm{d}}{V},\end{eqnarray} \tag{ 21 }$$

with

$$\begin{eqnarray}&&{X}_{1}(r,t)=\displaystyle \frac{1}{2}{\chi }_{i,j}^{1}{E}_{i}{E}_{j}\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{X}_{2}(r,t)=\displaystyle \frac{1}{3}{\chi }_{i,j,k}^{2}{E}_{i}{E}_{j}{E}_{k},\end{eqnarray} \tag{ 23 }$$

where the summation on repeated indexes is understood and where here the interaction extends over the volume V of the non-linear medium. The last expression represents the non-linear interaction involving three electric fields and is responsible for fundamental optical non-linear processes like sum- and difference-frequencies generation, second hardmonic generation and also SPDC. The corresponding interaction Hamiltonian is:

$$\begin{eqnarray}&&{H}_{I}(t)=\displaystyle \frac{1}{3}{\int }_{V}{\chi }_{i,j,k}^{2}{E}_{i}{E}_{j}{E}_{k}{\rm{d}}{V},\end{eqnarray} \tag{ 24 }$$

In SPDC, the non-linear effect is small and the probability that a pump photon is down converted into two emitted photons is very low. The pump is usually very intense and not significantly depleted by the interaction, thus can be treated classically, whereas the quantum description of the down-converted fields is essential. They can be written as:

$$\begin{eqnarray}&&{\hat{E}}_{j}({\boldsymbol{r}},t)\propto \int [{\hat{a}}_{{{\boldsymbol{k}}}_{j}}{{\rm{e}}}^{{\rm{i}}({{\boldsymbol{k}}}_{j}{\boldsymbol{r}}-{\omega }_{j}t)}+{\rm{H}}.{\rm{C}}.]{{\rm{d}}}^{3}{{\boldsymbol{k}}}_{j},\end{eqnarray} \tag{ 25 }$$

where j = 1, 2. The electric field of a classical monochromatic pump propagating along the Z-axis direction is:

$$\begin{eqnarray}&&{E}_{{\rm{p}}}({\boldsymbol{r}},t)={A}_{{\rm{p}}}({\boldsymbol{\rho }}){{\rm{e}}}^{{\rm{i}}({{\boldsymbol{k}}}_{{\rm{p}}}z-{\omega }_{{\rm{p}}}t)}\end{eqnarray} \tag{ 26 }$$

where ${\boldsymbol{\rho }}$ is the coordinate vector in the transverse X–Y plane. Considering each wave vector divided into the longitudinal component (pump direction), k_jz, and transverse component, ${{\boldsymbol{q}}}_{j}$, the interaction Hamiltonian becomes:

$$\begin{eqnarray}{H}_{{\rm{I}}}(t) & \propto & \displaystyle \int {\chi }^{(2)}{A}_{p}({\boldsymbol{\rho }}){{\rm{e}}}^{{\rm{i}}({k}_{p}-{k}_{1,z}-{k}_{2,z})z}{{\rm{e}}}^{{\rm{i}}({{\boldsymbol{q}}}_{1}+{{\boldsymbol{q}}}_{2}){\boldsymbol{\rho }}}{{\rm{e}}}^{-{\rm{i}}({\omega }_{p}-{\omega }_{1}-{\omega }_{2})t}\\ & & \times {\hat{a}}_{{\omega }_{1}{{\boldsymbol{q}}}_{1}}{\hat{a}}_{{\omega }_{2}{{\boldsymbol{q}}}_{2}}{\rm{d}}{\omega }_{1}{\rm{d}}{\omega }_{2}{\rm{d}}{{\bf{q}}}_{1}{\rm{d}}{{\bf{q}}}_{2}{\rm{d}}{\boldsymbol{\rho }}{\rm{d}}{z}.\end{eqnarray} \tag{ 27 }$$

Initially the down-converted fields are in the vacuum state, and upon the interaction, the evolved state in the Schrödinger picture follows:

$$\begin{eqnarray}&&| \psi \rangle =\hat{S}| 0\rangle =\exp \left[-\displaystyle \frac{1}{{\rm{i}}{\hslash }}\int {H}_{I}({t}^{{\prime} }){{\rm{d}}{t}}^{{\prime} }\right]| 0\rangle \end{eqnarray} \tag{ 28 }$$

Considering L the length of the crystal, the integral along the z direction results in:

$$\begin{eqnarray}&&{\int }_{0}^{L}{{\rm{e}}}^{{\rm{i}}({k}_{{\rm{p}}}-{k}_{1z}-{k}_{2z})z}{\rm{d}}{z}={{L}{\rm{e}}}^{{\rm{i}}{\rm{\Delta }}{kz}/2}\mathrm{sinc}({\rm{\Delta }}{kL}/2),\end{eqnarray} \tag{ 29 }$$

where ${\rm{\Delta }}k={k}_{{\rm{p}}}-{k}_{1z}-{k}_{2z}$ is the longitudinal phase mismatch. In the limit $L\to \infty$, the sinc function becomes a delta function and the integral term is different from zero for the perfect phase matching condition, i.e ${\rm{\Delta }}k=0$. In the realistic situation, the finite thickness of the crystal allows a certain phase mismatch, whose measure is given by the width of the sinc central peak, inversely proportional to the crystal length.

Similarly, the surface integral in the transverse direction ${\boldsymbol{\rho }}$ leads to the Fourier transform of the pump profile $A({\boldsymbol{\rho }})$. In the approximation of the plane wave, $A({\boldsymbol{\rho }})={A}_{0}$, we have

$$\begin{eqnarray}&&{\int }_{S}A({\boldsymbol{\rho }}){{\rm{e}}}^{{\rm{i}}({{\boldsymbol{q}}}_{1}+{{\boldsymbol{q}}}_{2}){\boldsymbol{\rho }}}{\rm{d}}{\boldsymbol{\rho }}={A}_{0}\delta ({{\boldsymbol{q}}}_{1}+{{\boldsymbol{q}}}_{2}).\end{eqnarray} \tag{ 30 }$$

In this approximation, the down-converted modes are perfectly correlated in the transverse direction, i.e. the mode with transverse momentum ${\boldsymbol{q}}$ is correlated to a mode with momentum ($-{\boldsymbol{q}}$).

The integral over the interaction time of equation (28) leads to:

$$\begin{eqnarray}&&\int {{\rm{e}}}^{-{\rm{i}}({\omega }_{{\rm{p}}}-{\omega }_{1}-{\omega }_{2})t}{\rm{d}}{t}=\delta ({\omega }_{1}+{\omega }_{2}-{\omega }_{{\rm{p}}}).\end{eqnarray} \tag{ 31 }$$

This allows us to express the frequencies as ${\omega }_{1}=\tfrac{{\omega }_{p}}{2}+{\rm{\Omega }}$ and ${\omega }_{2}=\tfrac{{\omega }_{p}}{2}-{\rm{\Omega }}$, where $\tfrac{{\omega }_{{\rm{p}}}}{2}$ is the degenerate frequency. With this simplification, the evolution operator becomes:

$$\begin{eqnarray}&&\hat{S}=\exp \left[\int (f({\boldsymbol{q}},{\rm{\Omega }}){\hat{a}}_{{\boldsymbol{q}},{\rm{\Omega }}}^{\dagger }{\hat{a}}_{-{\boldsymbol{q}},-{\rm{\Omega }}}^{\dagger }-{\rm{H}}.{\rm{C}}.){{\rm{d}}}^{2}{\bf{q}}{\rm{d}}{\rm{\Omega }}\right]\end{eqnarray} \tag{ 32 }$$

where the phase matching function $f({\boldsymbol{q}},{\rm{\Omega }})$, contains information about the strength of interaction (proportional to the length of the non-linear medium and the pump amplitude) and the spatio-temporal bandwidth of the down-converted fields:

$$\begin{eqnarray*}&&f({\boldsymbol{q}},{\rm{\Omega }})={\chi }^{(2)}{A}_{0}{{L}{\rm{e}}}^{{\rm{i}}{\rm{\Delta }}z/2}\mathrm{sinc}\left(\displaystyle \frac{{\rm{\Delta }}k({\boldsymbol{q}},{\rm{\Omega }})L}{2}\right).\end{eqnarray*}$$

The quantum state of SPDC modes at the start of the process is a vacuum state and due to the time evolution becomes:

$$\begin{eqnarray}&&| \psi \rangle =\exp \left[\int (f({\boldsymbol{q}},{\rm{\Omega }}){\hat{a}}_{{\boldsymbol{q}},{\rm{\Omega }}}^{\dagger }{\hat{a}}_{-{\boldsymbol{q}},-{\rm{\Omega }}}^{\dagger }-{\rm{H}}.{\rm{C}}.){{\rm{d}}}^{2}{\bf{q}}{\rm{d}}{\rm{\Omega }}\right]| 0\rangle .\end{eqnarray} \tag{ 33 }$$

Considering discrete values of ${\bf{q}}$, Ω, the integral can be replaced by the summation:

$$\begin{eqnarray}&&| \psi \rangle =\exp \left[\displaystyle \sum _{{\boldsymbol{q}},{\rm{\Omega }}}f({\boldsymbol{q}},{\rm{\Omega }}){\hat{a}}_{{\boldsymbol{q}},{\rm{\Omega }}}^{\dagger }{\hat{a}}_{-{\boldsymbol{q}},-{\rm{\Omega }}}^{\dagger }-{\rm{H}}.{\rm{C}}.\right]| 0\rangle .\end{eqnarray} \tag{ 34 }$$

Since the operators appearing in equation (34) corresponding to different pairs of modes $({\boldsymbol{q}},{\rm{\Omega }})\ne ({\boldsymbol{q}}^{\prime} ,{\rm{\Omega }}^{\prime} )$ commute with each other, following the Baker–Campbell–Hausdorff formula, i.e ${{\rm{e}}}^{x(\hat{A}+\hat{B})}={{\rm{e}}}^{x\hat{A}}\cdot {{\rm{e}}}^{x\hat{B}}$ for $[\hat{A},\hat{B}]=0$, the above state can be written in the direct product form as follows:

$$\begin{eqnarray}&&| \psi \rangle =\mathop{\displaystyle \bigotimes }\limits_{{\boldsymbol{q}},{\rm{\Omega }}}\hat{S}({\boldsymbol{q}},{\rm{\Omega }})| 0\rangle \\ &&\,=\mathop{\displaystyle \,\bigotimes }\limits_{{\boldsymbol{q}},{\rm{\Omega }}}\exp [f({\boldsymbol{q}},{\rm{\Omega }}){\hat{a}}_{{\boldsymbol{q}},{\rm{\Omega }}}^{\dagger }{\hat{a}}_{-{\boldsymbol{q}},-{\rm{\Omega }}}^{\dagger }-{\rm{H}}.{\rm{C}}.]| 0\rangle .\end{eqnarray} \tag{ 35 }$$

In the plane-wave pump approximation, SPDC can be seen as a collection of independent states, each one involving two-modes with correlated transverse momenta and frequencies. Expanding the exponential it is possible to rewrite the state as a product of two-mode entangled states in the photon number (multimode TWB) [120]:

$$\begin{eqnarray}&&| \psi \rangle =\mathop{\displaystyle \bigotimes }\limits_{{\boldsymbol{q}},{\rm{\Omega }}}| {TWB}{\rangle }_{{\boldsymbol{q}},{\rm{\Omega }}}=\mathop{\displaystyle \bigotimes }\limits_{{\boldsymbol{q}},{\rm{\Omega }}}\displaystyle \sum _{n}{c}_{{\boldsymbol{q}},{\rm{\Omega }}}(n)| n{\rangle }_{{\boldsymbol{q}},{\rm{\Omega }}}| n{\rangle }_{-{\boldsymbol{q}},-{\rm{\Omega }}}\end{eqnarray} \tag{ 36 }$$

where the probability amplitude ${c}_{{\boldsymbol{q}},{\rm{\Omega }}}(n)\propto \sqrt{{\mu }^{n}/{(\mu +1)}^{n+1}}$ is a coefficient that can be considered constant and is related to the mean number of photons in the mode $({\boldsymbol{q}},{\rm{\Omega }})$, $\mu \,={\sinh }^{2}| f({\boldsymbol{q}},{\rm{\Omega }})|$.

We are now interested in the statistical distribution of photons for a couple of conjugated modes, indicated by ${\hat{a}}_{({\boldsymbol{q}},{\rm{\Omega }})}\to {\hat{a}}_{1}$ and ${\hat{a}}_{(-{\boldsymbol{q}},-{\rm{\Omega }})}\to {\hat{a}}_{2}$. To calculate this it is convenient to consider one of the evolution operators in equation (35) (the so-called two-mode squeezing operator):

$$\begin{eqnarray}&&{\hat{S}}_{\mathrm{1,2}}=\exp [f({\boldsymbol{q}},{\rm{\Omega }}){\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}^{\dagger }-{\rm{H}}.{\rm{C}}.]\end{eqnarray} \tag{ 37 }$$

acting only on conjugated modes. For simplicity, we rewrite the complex amplitude as $f({\boldsymbol{q}},{\rm{\Omega }})={{r}{\rm{e}}}^{{\rm{i}}\theta }$ where $r({\boldsymbol{q}},{\rm{\Omega }})\,={A}_{0}L\ \mathrm{sinc}\left(\tfrac{{\rm{\Delta }}k({\boldsymbol{q}},{\rm{\Omega }})L}{2}\right)$ and $\theta ({\boldsymbol{q}},{\rm{\Omega }})={\rm{\Delta }}k({\boldsymbol{q}},{\rm{\Omega }})z/2$. The real quantity r is usually referred to as the squeezing parameter. The input–output relations for mode 1 and mode 2 follows as [120]:

$$\begin{eqnarray}&&{\hat{S}}_{1,2}^{\dagger }{\hat{a}}_{1}{\hat{S}}_{\mathrm{1,2}}={U}_{1}{\hat{a}}_{1}+{V}_{1}{\hat{a}}_{2}^{\dagger }\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{\hat{S}}_{1,2}^{\dagger }{\hat{a}}_{2}{\hat{S}}_{\mathrm{1,2}}={U}_{2}{\hat{a}}_{2}+{V}_{2}{\hat{a}}_{1}^{\dagger },\end{eqnarray} \tag{ 39 }$$

where:

$$\begin{eqnarray}&&{U}_{1}={U}_{2}=\cosh (r),\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{V}_{1}={V}_{2}={{\rm{e}}}^{{\rm{i}}\theta }\sinh (r).\end{eqnarray} \tag{ 41 }$$

Now we are able to calculate the mean photon number for the mode j (j = 1, 2):

$$\begin{eqnarray}\mu =\langle {\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}\rangle & = & \langle 0,0| {\hat{S}}^{\dagger }{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}\hat{S}| 0,0\rangle \\ & = & \langle 0,0| {\hat{S}}^{\dagger }{\hat{a}}_{j}^{\dagger }\hat{S}{\hat{S}}^{\dagger }{\hat{a}}_{j}\hat{S}| 0,0\rangle \\ & = & \langle 0,0| [{\hat{a}}_{1}^{\dagger }\cosh (r)+{\hat{a}}_{2}\sinh (r){{\rm{e}}}^{-{\rm{i}}\theta }]\\ & & \times [{\hat{a}}_{1}\cosh (r)+{\hat{a}}_{2}^{\dagger }\sinh (r){{\rm{e}}}^{{\rm{i}}\theta }]| 0,0\rangle \\ & = & {\sinh }^{2}(r).\end{eqnarray} \tag{ 42 }$$

where we have used the unitary condition ${\hat{S}}^{\dagger }\hat{S}=1$.

It is possible to derive the statistical momenta of superior orders by following the same steps as in the previous calculation. In particular we are interested in the second order moments (normally ordered):

$$\begin{eqnarray}\langle :{\hat{n}}_{1}{\hat{n}}_{2}:\,\rangle & = & \langle {\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}^{\dagger }{\hat{a}}_{1}{\hat{a}}_{2}\rangle ={\sinh }^{2}(r){\cosh }^{2}(r)+{\sinh }^{4}(r)\\ & = & 2{\mu }^{2}+\mu .\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&\langle :{\hat{n}}_{1}{\hat{n}}_{1}:\,\rangle =\langle :{\hat{n}}_{2}{\hat{n}}_{2}:\rangle =2{\sinh }^{4}(r)=2{\mu }^{2}\end{eqnarray} \tag{ 44 }$$

and in the variance of single modes and their covariance:

$$\begin{eqnarray}\langle {({\rm{\Delta }}{\hat{n}}_{1})}^{2}\rangle & = & \langle :{\hat{n}}_{1}{\hat{n}}_{1}:\,\rangle -\langle {\hat{n}}_{1}{\rangle }^{2}+\langle {\hat{n}}_{1}\rangle \\ & = & \langle {\hat{n}}_{1}\rangle (1+\langle {\hat{n}}_{1}\rangle )=\mu (1+\mu )=\langle {({\rm{\Delta }}{\hat{n}}_{2})}^{2}\rangle \end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}\langle :{\rm{\Delta }}{\hat{n}}_{1}{\rm{\Delta }}{\hat{n}}_{2}:\,\rangle & = & \langle :{\hat{n}}_{1}{\hat{n}}_{2}:\,\rangle -\langle {\hat{n}}_{1}\rangle \langle {\hat{n}}_{2}\rangle \\ & = & \langle {\hat{n}}_{1}\rangle (1+\langle {\hat{n}}_{1}\rangle )=\langle {\hat{n}}_{2}\rangle (1+\langle {\hat{n}}_{2}\rangle )=\mu (1+\mu ).\end{eqnarray} \tag{ 46 }$$

From equation (45) it can be seen that the single mode of the SPDC radiation has thermal statistics with a super-poissonian component equal to ${\mu }^{2}$ (excess noise).

In the previous paragraph we derived the statistical behavior of SPDC photons emitted in two conjugated modes. Here we are interested in the statistical behavior of the detected photons $\langle {\hat{N}}_{j}\rangle$.

According to the simple detection model of section 2, equations (3), and the results of section 4.2, it easily follows that:

$$\begin{eqnarray}&&\langle {\hat{N}}_{j}\rangle ={\eta }_{j}\langle {\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}\rangle ={\eta }_{j}\mu ,\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&\langle {({\rm{\Delta }}{\hat{N}}_{j})}^{2}\rangle ={\eta }_{j}^{2}{\mu }^{2}+{\eta }_{j}\mu ,\end{eqnarray} \tag{ 48 }$$

and the measured covariance is:

$$\begin{eqnarray}&&\langle :{\rm{\Delta }}{\hat{N}}_{1}{\rm{\Delta }}{\hat{N}}_{2}:\,\rangle ={\eta }_{1}{\eta }_{2}\mu (1+\mu ).\end{eqnarray} \tag{ 49 }$$

Now it is possible to calculate the parameters that quantify the degree of correlation between two modes and, in particular, the NRF σ defined in equation (12). As described in section 3.1, the NRF allows discrimination between classical states of light and quantum states of light. If $\sigma \geqslant 1,$ we have classical light such as thermal light or coherent light if $\sigma =1$, and if $\sigma \lt 1$ we have quantum correlated light.

Substituting the statistics of two conjugated modes of SPDC, equations (45) and (46), in the definition of equation (12) we can obtain perfect correlations in the ideal lossless case i.e.:

$$\begin{eqnarray}&&\sigma =0,\end{eqnarray} \tag{ 50 }$$

while when losses are considered, according to equation (13), we have:

$$\begin{eqnarray}&&{\sigma }_{{\rm{\det }}}\simeq 1-\eta \end{eqnarray} \tag{ 51 }$$

where we assume ${\eta }_{1}={\eta }_{2}=\eta$. For unbalanced losses, the NRF becomes:

$$\begin{eqnarray}&&{\sigma }_{{\rm{\det }}}=1-\bar{\eta }+\displaystyle \frac{{({\eta }_{1}-{\eta }_{2})}^{2}}{2\bar{\eta }}\left(\mu +\displaystyle \frac{1}{2}\right),\end{eqnarray} \tag{ 52 }$$

where μ is the mean number of photons per mode and $\bar{\eta }$ is the mean quantum efficiency. Equation (51) shows how the measured NRF between two conjugated modes is always smaller than 1 in the case of identical quantum efficiency. Otherwise, if we have ${\eta }_{1}\ne {\eta }_{2}$ there is an additional positive term, proportional to the mean value of photons per mode, which arise from a non-perfect cancellation of the excess noise of the thermal fluctuation. This can lead to the measure ${\sigma }_{{\rm{\det }}}\gt 1$ losing the non-classical signature, even in case of perfectly correlated quantum light.

These results are valid in the plane-wave pump approximation. In the experiments, the momentum distribution of the pump which cannot be a delta function, generates an uncertainty in the relative momentum (direction of propagation) of correlated photons. Therefore, a full study of the mode collection inside finite detection areas is needed for describing the experimental results and some issues relating to the detection of non-classical features of a multimode squeezed vacuum.

In the far field region, obtained at the focal plane of a thin lens in an f−f configuration, any transverse mode ${\boldsymbol{q}}$ is associated with a single position ${\boldsymbol{x}}$ according to the geometric transformation $(2{cf}/\omega ){\boldsymbol{q}}\to {\boldsymbol{x}}$, where c is the speed of light. The exact condition ${{\boldsymbol{q}}}_{1}+{{\boldsymbol{q}}}_{2}=0$ for correlated photons, which comes from the integral in equation (30) in the plane wave pump approximation, becomes in the far field a strict condition on their positions, ${{\boldsymbol{x}}}_{1}+{{\boldsymbol{x}}}_{2}=0$. For degenerate frequencies, ${\omega }_{1}={\omega }_{2}={\omega }_{p}/2$, correlated photons reach symmetric positions with respect to the pump intersection point (${\boldsymbol{x}}=0$). A more realistic Gaussian distributed pump with angular spread ${\rm{\Delta }}{\boldsymbol{q}}$ leads to uncertainty on the position of the correlated photon, ${{\boldsymbol{x}}}_{1}+{{\boldsymbol{x}}}_{2}=0\pm {\rm{\Delta }}{\boldsymbol{x}}$, where ${\rm{\Delta }}{\boldsymbol{x}}\,=(2{cf}/{\omega }_{p}){\rm{\Delta }}{\boldsymbol{q}}$ represents the size, in the far field, of the coherence area ${{ \mathcal A }}_{{\rm{coh}}}$ in which it is possible to collect photons from correlated modes. It is possible to visualize coherence areas in the high gain regime ($r\gt 1$) where they appear as correlated spots (speckles) around symmetrical positions ${\boldsymbol{x}}$ and $-{\boldsymbol{x}}$, where the center of symmetry (CS) is basically the pump-detection plane interception. These correlations in photon numbers can be seen in figure 3.

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It is possible to measure the size of this coherence area by performing the spatial cross correlation between the two beams:

$$\begin{eqnarray*}&&c(\xi )=\displaystyle \sum _{{\bf{x}}}\displaystyle \frac{\langle \delta {\hat{N}}_{1}({\bf{x}})\delta {\hat{N}}_{2}(-{\bf{x}}+\xi )\rangle }{\sqrt{\langle {[\delta {\hat{N}}_{1}({\bf{x}})]}^{2}\rangle \langle {[\delta {\hat{N}}_{2}(-{\bf{x}}+\xi )]}^{2}\rangle }}\end{eqnarray*}$$

where $\xi =(x,y)$ is the shift.

In order to collect most of the correlated photons, two symmetrically placed detectors must have sensitive areas larger than the coherence area A_coh. Referring to figure 4, we consider two equal and symmetric areas ${{ \mathcal A }}_{{\rm{\det }},j}$ $(j=1,2)$ containing a large number of transverse spatial modes ${{ \mathcal M }}_{c}={A}_{{\rm{\det }},j}/{A}_{{\rm{coh}}}$, represented by the light blue circles in the figure. However, there are modes ${{ \mathcal M }}_{{\rm{b}}}$ on the detector border that are only partially detected, namely with efficiency β, which can be assumed to be equal to 1/2 on average. Moreover, experimental misalignment, δ, can lead to the collection of some uncorrelated modes ${{ \mathcal M }}_{{\rm{u}}}$. Even if it is possible to optimize the experiment in order to reduce the contributions of ${{ \mathcal M }}_{{\rm{b}}}$ and ${{ \mathcal M }}_{{\rm{u}}}$, it is necessary to take them into account for a complete description of the physical scenario.

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Since each SPDC couple of modes is independent of the others, the variance and covariance of a state with ${ \mathcal M }$ pairs are ${ \mathcal M }$ times the values of a single pair. Therefore, taking into account the contribution of the different kinds of involved modes and the single/two-mode statistics in equations (47), (48), and (49) one has:

$$\begin{eqnarray}&&\langle {\hat{N}}_{j}\rangle =({{ \mathcal M }}_{{\rm{c}}}+{{ \mathcal M }}_{{\rm{u}}}+{{ \mathcal M }}_{{\rm{b}}}\beta ){\eta }_{j}\mu \end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}\langle {(\delta {\hat{N}}_{j})}^{2}\rangle & = & ({{ \mathcal M }}_{{\rm{c}}}+{{ \mathcal M }}_{{\rm{u}}}){\eta }_{j}\mu (1+{\eta }_{j}\mu )\\ & & +{{ \mathcal M }}_{{\rm{b}}}\beta {\eta }_{j}\mu (1+\beta {\eta }_{j}\mu )\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&\langle \delta {\hat{N}}_{1}\delta {\hat{N}}_{2}\rangle =({{ \mathcal M }}_{{\rm{c}}}+{{ \mathcal M }}_{{\rm{b}}}{\beta }^{2}){\eta }_{1}{\eta }_{2}\mu (1+\mu ).\end{eqnarray} \tag{ 55 }$$

where μ is the mean photon number per mode and ${\eta }_{1}$ and ${\eta }_{2}$ are the detection efficiencies of the two channels.

Substituting the previous expressions into the definition of the NRF in equation (12) we have (${\eta }_{1}={\eta }_{2}=\eta$):

$$\begin{eqnarray}&&{\sigma }_{{\rm{\det }}}\simeq 1-\eta A.\end{eqnarray} \tag{ 56 }$$

The quantity $0\lt A\lt 1$ can be interpreted as the collection efficiency of correlated photons pairs (or modes) and it assumes the form:

$$\begin{eqnarray}&&A=({{ \mathcal M }}_{{\rm{c}}}+{{ \mathcal M }}_{{\rm{b}}}{\beta }^{2}-{{ \mathcal M }}_{{\rm{u}}}\mu )/({{ \mathcal M }}_{{\rm{c}}}+{{ \mathcal M }}_{{\rm{u}}}+{{ \mathcal M }}_{{\rm{b}}}\beta )\end{eqnarray} \tag{ 57 }$$

It is possible to evaluate this collection efficiency using just some basic geometrical considerations: in figure 4 we call δ the misalignment, r the coherence radius at the detection plane (the radius of the light blue circles) and L the linear size of a detection region. Under the conditions $L\gt 2r$ and $\delta \ll L$, the counts of the different types of modes are related to the geometrical measurable parameters as:

$$\begin{eqnarray}&&{{ \mathcal M }}_{{\rm{u}}}=2L\delta /\pi {r}^{2},\end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray}&&{{ \mathcal M }}_{{\rm{c}}}=[{(L-2r)}^{2}-2L\delta ]/\pi {r}^{2},\end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray}&&{{ \mathcal M }}_{{\rm{b}}}=2L/r.\end{eqnarray} \tag{ 60 }$$

By introducing the dimensionless parameters $X=L/2r$ and $D=\delta /2r$, the collection efficiency becomes:

$$\begin{eqnarray}&&A=\displaystyle \frac{X(\pi {\beta }^{2}-2D(\mu +1)-2)+{X}^{2}+1}{{X}^{2}+(\pi \beta -2)X+1}.\end{eqnarray} \tag{ 61 }$$

Thus, in the limit $\mu \to 0$, the measured NRF in equation (56) does not depend on the mean number of photons. In the asymptotic limit $X\gg 1$, i.e. when the detection size is much larger than the correlation area, A approaches unity and the NRF reaches the value in equation (51), the one of two correlated modes in the monochromatic plane-wave pump approximation.

In wide field absorption imaging the sample is illuminated by a probe state and the transmitted pattern is detected by the pixels of a 2D matrix, e.g. a CCD camera. The absorption coefficient α in a point of the sample, is estimated by the measurement of the photon number $\langle \hat{N}\rangle$ detected by a corresponding pixel [131]. The uncertainty is:

$$\begin{eqnarray}&&{\rm{\Delta }}\alpha =\displaystyle \frac{\sqrt{{{\rm{\Delta }}}^{2}\langle \hat{N}\rangle }}{\left|\tfrac{\partial \langle N\rangle }{\partial \alpha }\right|}.\end{eqnarray} \tag{ 62 }$$

In the following we will consider the uncertainty ${\rm{\Delta }}\alpha$ in two different measurement schemes, referred as to direct (DR) and differential (DIFF) imaging, respectively.

The DR imaging scheme is represented in figure 5(a). A single probe beam is addressed to the object and the transmitted part is collected by the detector.

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The losses due to the sample can be modeled by a BS with transmission coefficient $1-\alpha$. Referring to equation (3) of section 2 and substituting η with $1-\alpha$, the mean detected photon number is $\langle \hat{N}\rangle =(1-\alpha )\langle \hat{n}\rangle$, where $\langle \hat{n}\rangle$ is the mean number of the detected photons, as it would be without the object. The variance becomes:

$$\begin{eqnarray}&&\langle {{\rm{\Delta }}}^{2}\hat{N}\rangle ={(1-\alpha )}^{2}[\langle {{\rm{\Delta }}}^{2}\hat{n}\rangle -\langle \hat{n}\rangle ]+(1-\alpha )\langle \hat{n}\rangle .\end{eqnarray} \tag{ 63 }$$

Substituting equation (63) in equation (62), the uncertainty in the absorption estimation for the DR imaging scheme is

$$\begin{eqnarray}&&{\rm{\Delta }}{\alpha }_{{\rm{DR}}}=\sqrt{\displaystyle \frac{{(1-\alpha )}^{2}[F-1]+(1-\alpha )}{\langle \hat{n}\rangle }},\end{eqnarray} \tag{ 64 }$$

where F is the Fano factor defined in section 3, as it would be measured in absence of the object. For a classical probe state (lower bounded by F = 1), the sensitivity scales as ${\rm{\Delta }}{\alpha }_{{\rm{DR}}}=\sqrt{(1-\alpha )/\langle \hat{n}\rangle }$ which represents the shot-noise limit. Furthermore, from equation (64), it is evident that a probe state with non-classical statistics, i.e. a value of F smaller than the unit, allows going beyond the shot-noise limit. It can be demonstrated that a Fock state $| n\rangle$, with F = 0, allows one to reach the ultimate quantum limit in the precision of absorption estimation. A Fock state with n = 1 can be approximated experimentally by a heralding single-photon source and it has been used recently in an absorption spectroscopy experiment [127]. However, as discussed in section 3, the Fano factor is deteriorated by the detection loss η. Thus, the non-classical behavior in terms of noise reduction is lower bounded by ${F}_{{\rm{\det }}}=1-\eta$. It is important to note that splitting a single-mode beam into ${ \mathcal N }$ pixels leads to a detection probability of the order of $\eta \leqslant 1/{ \mathcal N }$ for each of them, ruling out the possibility of using a single mode for reaching SSN sensitivity in WFI. Even if sub-Poissonian light ($F\lt 1$) in single mode or few modes have been obtained, experimental complications in their generation and simultaneous detection limit their use for imaging, where higher number of non-classical spatial modes are needed, each mode addressing a single pixel. On the other hand, as we showed in section 4, the SPDC process naturally produces a pair of beams, which are (individually) spatially incoherent (containing thousands of independent spatial modes) but are locally correlated in the photon number. Even if fluctuations of a single spatial mode in one beam are super-poissonian, due to photon-number entanglement these fluctuations are perfectly replicated in the correlated mode of the second beam. This property can be applied in a DIFF imaging scheme as first proposed in [129] and described in the following.

Differential imaging exploits the correlation properties of two beams instead of one. These can be, for example, TWBs generated by SPDC as represented in figure 5(c), or a thermal beam split by a 50:50 BS, depicted in figure 5(b). The scheme is the following: one of the two beams impinges on an absorbing object, with transmittance ($1-\alpha$), before being detected. The other beam is directly detected, playing the role of reference for the noise. Assuming for simplicity the same detection losses along the two optical paths, the difference in mean photon numbers is proportional to the absorption coefficient:

$$\begin{eqnarray}&&\langle {\hat{N}}_{-}\rangle =\langle {\hat{N}}_{2}-{\hat{N}}_{1}\rangle =\alpha \langle \hat{n}\rangle .\end{eqnarray} \tag{ 65 }$$

The variance in the photon-number difference can be expressed in terms of the Fano factor and the NRF, σ, defined in equation (12), in the absence of the object, as:

$$\begin{eqnarray}&&\langle {{\rm{\Delta }}}^{2}{\hat{N}}_{-}\rangle =[{\alpha }^{2}(F-1)+\alpha +2\sigma (1-\alpha )]\langle \hat{n}\rangle .\end{eqnarray} \tag{ 66 }$$

Therefore, the sensitivity in a DIFF scheme can be evaluated according to equation (62), where $N\to {N}_{-}$, as:

$$\begin{eqnarray}&&{\rm{\Delta }}{\alpha }_{{\rm{DIFF}}}=\sqrt{\displaystyle \frac{{\alpha }^{2}(F-1)+\alpha +2\sigma (1-\alpha )}{\langle \hat{n}\rangle }}.\end{eqnarray} \tag{ 67 }$$

The expression of the differential classical (DC) scheme can be obtained from equation (67) by substituting $\sigma =1$. For a weakly absorbing object, $\alpha \to 0$, the term ${\alpha }^{2}(F-1)$ is very small even for super-Poissonian sources and can be neglected. Thus, the uncertainty in the DC scheme becomes ${\rm{\Delta }}{\alpha }_{{\rm{DC}}}=\sqrt{(2-\alpha )/\langle \hat{n}\rangle }$, which is a factor of $\sqrt{2}$ larger than the DR imaging for small α. The uncertainty achieved by the quantum correlations with $\sigma \lt 1$, namely ${\rm{\Delta }}{\alpha }_{{\rm{SSN}}}\,={\rm{\Delta }}{\alpha }_{{\rm{DIFF}}}(\sigma \lt 1)$ can be compared to both the direct and the classical DIFF imaging by using equation (64) and equation (67) in the relevant limits discussed before:

$$\begin{eqnarray}\displaystyle \frac{{\rm{\Delta }}{\alpha }_{{\rm{SSN}}}}{{\rm{\Delta }}{\alpha }_{{\rm{DC}}}} & = & \displaystyle \frac{{{SNR}}_{{\rm{DC}}}}{{{SNR}}_{{\rm{SSN}}}}=\sqrt{\displaystyle \frac{\alpha +2\sigma (1-\alpha )}{2-\alpha }}\approx \sqrt{\sigma }\\ \displaystyle \frac{{\rm{\Delta }}{\alpha }_{{\rm{SSN}}}}{{\rm{\Delta }}{\alpha }_{{\rm{DR}}}} & = & \displaystyle \frac{{{SNR}}_{{\rm{DR}}}}{{{SNR}}_{{\rm{SSN}}}}=\sqrt{\displaystyle \frac{\alpha +2\sigma (1-\alpha )}{1-\alpha }}\approx \sqrt{2\sigma }.\end{eqnarray} \tag{ 68 }$$

Here, we have introduced the signal-to-noise ratio (SNR), ${SNR}=\alpha /{\rm{\Delta }}\alpha$, as an equivalent figure of merit of the measured sensitivity. From equation (68), it is clear that the advantage of quantum correlation can be quantified by the value of the non-classical parameter σ, which for the TWB is lower bounded only by the loss factor $\sigma =1-\eta$ (see section 4.3). In particular, the SSN condition, $\sigma \lt 1$, guarantees an advantage with respect to the DC scheme, while a more restrictive condition, $\sigma \lt 1/2$, is needed for the SSN scheme to beat the direct (shot-noise limited) one. This condition corresponds to the requirement of an overall loss in the detection of correlated photons smaller than 50%.

In fact one of the difficulties of the technique, when addressed to SSNWFI, is to achieve a good collection efficiency of the correlated modes in the far field without sacrificing the spatial resolution. This is due to the trade-off between the collection efficiency and the pixel size, as discussed in section 4.4.

The first experimental demonstration of the SSNWFI involving many spatial modes was given in 2010 [130]. Figure 6 presents the advantages of quantum DIFF imaging over DR and DC schemes. The weak absorbing 'π'-shaped object is hidden in the noise for both the classical imaging techniques, whereas its shape can be clearly identified in the image obtained using the SSN schemes of figure 5(c).

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It is important to mention that, in the experiment discussed in [130], the image was obtained without any imaging lenses, basically revealing the shadow of the object placed closed to the detection plane. Thus, the resolution was not high enough for any potential application in the real world, especially in microscopy, where the technique would be naturally addressed. Moreover, the average NRF achieved was just slightly below 0.5, enough to surpass DC imaging but not sufficient to provide a real exploitable advantage with respect to DR imaging in realistic conditions.

Very recently, an important step forward was made with the realization of SSNWFI in a real microscopic configuration [53]. A noise reduction such as $\sigma =0.8$ was obtained for each pixel in a matrix of approximately 8000 pixels, and a spatial resolution of 5 μ m at the sample. This noise reduction is enough to beat DC imaging, and the resolution is sufficient for the imaging of complex structures, such as cells. Reducing the resolution by one third, allows the performance of the DR imaging scheme to be easily overcome. The trade-off between the noise reduction, represented by the value of the paramater $\sigma$, and the resolution, according to the model of the collection efficiency developed in section 4.4, is reported in figure 7. The corresponding improvement of the SNR with respect to both differential and direct shot-noise limited classical imaging schemes. The main difficulties in comparison to classical microscopy are that the imaging systems should be able to reduce the aberration without introducing any losses. Figure 8 shows the experimental image profile of the sample (a 'ϕ'-shaped few-nanometer-thick deposition with absorption coefficient $\alpha =1 \%$) at different resolution scale L ($L=d\cdot 5\,\mu {\rm{m}}$). From $L=15\mu$ (d = 3) the object starts to appear in the SSN image, while remains almost undefined in the classical images.

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Finally, we mention that, different from previous proofs of principle of quantum enhanced phase-contrast microscopy exploiting NOON states (with N = 2) [23–26], the SSN wide-field microscope can offer the possibility of dynamic imaging without scanning the whole sample.

Let us consider first the direct scheme in which a probe with mean photon number n is addressed to the object and its reflected (transmitted) part reaches a detector together with a much stronger background with mean photon number n_B. Note that in this case only one beam is used. In hypothesis H₀ only the background reaches the detector and the measured photon number is n_B, while for H₁ one has ${\eta }_{{\rm{P}}}n+{n}_{{\rm{B}}}$, their difference being the signal. The variance of the measurement is assumed to be dominated in both cases by the background fluctuations $\langle {\delta }^{2}{\hat{n}}_{{\rm{B}}}\rangle$. Thus, the SNR obtained by ${ \mathcal K }$ measurements is:

$$\begin{eqnarray}&&{{SNR}}_{{\rm{Dr}}}=\displaystyle \frac{\sqrt{{ \mathcal K }}{\eta }_{{\rm{P}}}n}{\sqrt{2\langle {\delta }^{2}{\hat{n}}_{{\rm{B}}}}\rangle }.\end{eqnarray} \tag{ 69 }$$

Even if the strategy described above seems the simplest and the most natural approach, it assumes implicitly that it is possible to have a separate estimation of the mean photon number of the background, for example by a measurement made in the absence of the object. If the background cannot be measured separately, for example because the object is constantly present (of course this information is not available a priori), the previous method, simply based on the discrimination of two average intensity levels cannot be applied. A second-order measurement of the intensity is required instead, therefore we consider $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}\rangle$, where $\delta {\hat{n}}_{1}$ and $\delta {\hat{n}}_{2}$ are the fluctuations on the reference and 'probe+noise' beams, respectively. In the absence of the target, the background and reference are uncorrelated, thus the covariance $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{H}_{0}}$ is null and it establishes the natural zero-offset for the measurement. In the presence of the target $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{H}_{1}}$ is, in general, different from zero and depends on the exploited state, i.e. on the correlations in the photon-number fluctuations between the reference and the reflected probe beam. In order to calculate the SNR it is necessary to also consider the uncertainty of the covariance. The fluctuation of this quantity is, by definition, for i = 0, 1:

$$\begin{eqnarray}&&{\langle {\delta }^{2}(\delta {\hat{n}}_{1}\delta {\hat{n}}_{2})\rangle }_{{H}_{i}}\equiv {\langle {(\delta {\hat{n}}_{1}\delta {\hat{n}}_{2})}^{2}\rangle }_{{H}_{i}}-{\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}\rangle }_{{H}_{i}}^{2}.\end{eqnarray} \tag{ 70 }$$

As before, if we consider the fluctuation in n₂ dominated by the background, both in the presence and in the absence of the target, i.e. $\delta {n}_{2}{| }_{{H}_{1}}\approx \delta {n}_{2}{| }_{{H}_{0}}=\delta {n}_{B}$, it immediately follows that $\langle {\delta }^{2}(\delta {\hat{n}}_{1}\delta {\hat{n}}_{2})\rangle =\langle {\delta }^{2}{\hat{n}}_{1}\rangle \langle {\delta }^{2}{\hat{n}}_{{\rm{B}}}\rangle$. Exploiting the quantum correlation in photon-number fluctuations of the TWB state, according to section 4.3, $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{H}_{1}}\,=n{\eta }_{p}{\eta }_{R}(1+n/M)$, where M is the number of modes. Since each beam of the TWBs is multithermal (with M modes): $\langle {\delta }^{2}{\hat{n}}_{1}\rangle \,=n{\eta }_{{\rm{R}}}(1+{\eta }_{{\rm{R}}}n/M)$, the SNR can be evaluated as:

$$\begin{eqnarray}{{SNR}}_{{\rm{SPDC}}} & \approx & \displaystyle \frac{\sqrt{{ \mathcal K }}\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{H}_{1}}}{\sqrt{2\langle {\delta }^{2}{\hat{n}}_{1}\rangle \langle {\delta }^{2}{\hat{n}}_{{\rm{B}}}\rangle }}\\ & = & \displaystyle \frac{\sqrt{{ \mathcal K }}n{\eta }_{{\rm{P}}}{\eta }_{{\rm{R}}}(1+n/M)}{\sqrt{2n{\eta }_{{\rm{R}}}(1+{\eta }_{{\rm{R}}}n/M)\langle {\delta }^{2}{\hat{n}}_{{\rm{B}}}\rangle }}\end{eqnarray} \tag{ 71 }$$

$$\begin{eqnarray}&&\approx \,\displaystyle \frac{\sqrt{{ \mathcal K }{\eta }_{R}n}{\eta }_{P}}{\sqrt{2\langle {\delta }^{2}{\hat{n}}_{B}\rangle }}\end{eqnarray} \tag{ 72 }$$

where the last approximation holds for $n/M\ll 1$, i.e. when the mean photon number per mode $n/M=\mu$ is small. We can now compare this result with the SNR obtained with the direct measurement of the probe mean photon number (when this is possible). This results in an improvement for $n\lt 1$ as large as ${{SNR}}_{{\rm{SPDC}}}/{{SNR}}_{{\rm{Dr}}}={({\eta }_{{\rm{R}}}/n)}^{1/2}$.

It is also interesting to evaluate the advantage of the quantum correlation with respect to the possible use of classically correlated states. The first line of equation (71) shows that the classical and quantum schemes with the same local statistics only differ for the strength of the correlation, quantified by the covariance $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{H}_{1}}$. According to the generalized Cauchy–Schwarz inequality presented in section 3, the covariance for classical beams is bounded by $\varepsilon =\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}\rangle \,/{(\langle :{\delta }^{2}{\hat{n}}_{1}:\rangle \langle :{\delta }^{2}{\hat{n}}_{2}:\rangle )}^{1/2}\leqslant 1$. Split thermal beams saturate the inequality, ${\varepsilon }_{{\rm{TH}}}=1$, with $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{\rm{TH}}}\,={\eta }_{{\rm{R}}}{\eta }_{{\rm{P}}}{n}^{2}/M$, thus representing the best classical strategy. On the other hand the SPDC quantum correlation provides ${\varepsilon }_{{\rm{SPDC}}}=M/n+1$ with $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}{\rangle }_{{\rm{SPDC}}}={\eta }_{{\rm{P}}}{\eta }_{{\rm{R}}}n(1+n/M)$. Therefore, the comparison of the SNR with classical and quantum correlation immediately gives:

$$\begin{eqnarray}&&\displaystyle \frac{{{SNR}}_{{\rm{SPDC}}}}{{{SNR}}_{{\rm{TH}}}}={\varepsilon }_{{\rm{SPDC}}}=\displaystyle \frac{M}{n}+1=\displaystyle \frac{1}{\mu }+1.\end{eqnarray} \tag{ 73 }$$

A dramatic quantum enhancement for a photon number per mode $\mu =n/M\ll 1$ is evident. It is important to notice that, as anticipated, the enhancement does not depend on the background intensity and it is also immune to the losses.

Finally we would like to trace a connection between the QI using the OPA receiver of [137] and the intensity-measurement-based scenario described above, showing that they have the same non-classicality/entanglement breaking condition. Indeed, for a zero-mean Gaussian distributed bipartite state, the moment-factoring theorem allows writing the photon-number covariance in terms of the modulus of the phase-sensitive cross correlation (which is the quantity measured by the OPA receiver in [137]): $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}\rangle =| \langle {\hat{a}}_{1}{\hat{a}}_{2}\rangle {| }^{2}$. On the other hand the normal ordered variance for a gaussian mode can be written in terms of the mean photon number: $\langle :{\delta }^{2}{\hat{n}}_{j}:\rangle =\langle {\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}{\rangle }^{2}$. Therefore, the non-classicality breaking condition, represented in general by the violation of the Cauchy–Schwarz inequality, in the framework of Gaussian states, coincides with the entanglement breaking condition $| \langle {\hat{a}}_{1}{\hat{a}}_{2}\rangle {| }^{2}\leqslant \langle {\hat{n}}_{1}\rangle \langle {\hat{n}}_{2}\rangle$ reported in [137], valid for two conjugated modes. Substituting in the Cauchy–Schwarz inequality the explicit expression of the photon statistics at the detectors, in the general multimode case, the condition becomes:

$$\begin{eqnarray}&&{\eta }_{{\rm{P}}}{\eta }_{{\rm{R}}}n\left(1+\displaystyle \frac{n}{M}\right)\leqslant {\left[{\eta }_{{\rm{R}}}^{2}\displaystyle \frac{{n}^{2}}{M}\left({\eta }_{{\rm{P}}}^{2}\displaystyle \frac{{n}^{2}}{M}+\displaystyle \frac{{n}_{{\rm{B}}}^{2}}{{M}_{{\rm{B}}}}\right)\right]}^{1/2}.\end{eqnarray} \tag{ 74 }$$

In the limit of $\mu =n/M\ll 1$ the condition simplifies as ${n}_{{\rm{B}}}\geqslant {\eta }_{{\rm{P}}}{({{MM}}_{{\rm{B}}})}^{1/2}$. For example when single modes are detected $M={M}_{{\rm{B}}}=1$, a mean number of background photons ${n}_{{\rm{B}}}\gt 1$ is enough to destroy Gaussian entanglement and more in general non-classical photon statistics; nevertheless the enhancement in the SNR remains.

The experimental setup used in [45] for the realization of the intensity-correlation-based QI protocol is represented in figure 11(a). Type II SPDC generates pairs of correlated 5 ns pulses with an average number of photons per spatio-temporal mode of $\mu \sim 0.1$, which are then addressed to a high quantum efficiency CCD camera. In the QI protocol (figure 11(a)) one beam (the reference) is directly detected, while a target object (a 50:50 BS) is placed in the path of the other one (the probe), where it is superimposed with a pseudothermal background produced by a laser beam scattered by an Arecchi rotating ground glass. When the object is removed, only the background reaches the detector. The CCD camera detects, in different regions, both the optical paths. In the classical illumination (CI) protocol (figure 11(b)), the TWBs are substituted with classical correlated beams, obtained by splitting a single arm of SPDC, that is a multithermal beam, and by adjusting the pump intensity to ensure the same local statistics and spatial coherence properties for the quantum and the classical source.

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In this scheme, n₁ and n₂ are the photon numbers detected by pairs of spatially correlated pixels in a single 5 ns shot of the pump laser, as presented in figures 11 (c)–(e). Since K = 80 correlated pixel pairs are present, it is possible to perform spatial statistics, which allows the evaluation of the covariance $\langle \delta {\hat{n}}_{1}\delta {\hat{n}}_{2}\rangle$ in a single shot, thus reducing the measurement time needed for asserting the presence or the absence of the target.

Figure 12 reports the measured ε versus the theoretical prediction. It can be seen that for the TWB ${\varepsilon }_{{\rm{QI}}}$ is in the quantum regime (${\varepsilon }_{{\rm{QI}}}\gt 1$) for small intensities of the thermal background, reaching the value ${\varepsilon }_{{\rm{QI}}}\simeq 10$ when ${n}_{{\rm{B}}}=0$. It rapidly decreases below the classical threshold according to the condition in equation (74) when the background increases. For classical correlation of split thermal beams, ${\varepsilon }_{{\rm{TH}}}$ is always in the classical regime, starting from ${\varepsilon }_{{\rm{TH}}}=1$ for ${n}_{{\rm{B}}}=0$, as expected.

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In figure 13 an experimental comparison of the SNR for quantum and classical illumination is presented. While the SNR unavoidably decreases when the noise increases for both QI and CI (see equation (71)), the ratio between them is constant regardless of the value of n_B, in agreement with the theoretical prediction provided in equation (71), ${{SNR}}_{{\rm{SPDC}}}/{{SNR}}_{{\rm{TH}}}={\varepsilon }_{{\rm{SPDC}}}\simeq 10$. In turn, the measurement time, i.e. the number of repetitions ${ \mathcal K }$ needed for discriminating the presence/absence of the target, is dramatically reduced by 100 times when quantum correlations are exploited.

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A scheme of a conventional GI technique experimental setup is shown in figure 14. The image of the object is retrieved by measuring a certain function $S({x}_{j})$, where x_j is the position of the pixel j of the resolving detector, in arm ${\text{}}1$. In general $S({x}_{j})$ involves the correlation functions of the output of the two detectors:

$$\begin{eqnarray}&&S({x}_{j})=f(E[{{\mathbb{N}}}_{2}],E[{N}_{1}({x}_{j})],E[{{\mathbb{N}}}_{2}{N}_{1}({x}_{j})],\ldots ,\,E[{{\mathbb{N}}}_{2}^{p}{N}_{1}^{q}({x}_{j})])\end{eqnarray} \tag{ 75 }$$

where ${{\mathbb{N}}}_{2}$ is the total number of photons collected at the bucket detector and ${N}_{1}({x}_{j})$ is the number of photons collected in the jth pixel of the resolving detector.

Experimentally these quantities are evaluated averaging the number of acquisitions ${ \mathcal K }$: $E[X]=\tfrac{1}{{ \mathcal K }}{\sum }_{k=1}^{{ \mathcal K }}{X}_{k}$.

The ghost image can be retrieved by exploiting different GI protocols, namely, different expressions for $S({x}_{j})$ [171]. We focus on the covariance between the two outputs (note that the covariance between the outputs has also been considered in the QI protocol based on intensity correlations described in section 6.1):

$$\begin{eqnarray}S({x}_{j}) & = & \mathrm{Cov}({{\mathbb{N}}}_{2},{N}_{1}({x}_{j}))\equiv E[\{{{\mathbb{N}}}_{2}-E[{{\mathbb{N}}}_{2}]\}\\ & & \times \{{N}_{1}({x}_{j})-E[{N}_{1}({x}_{j})]\}]\\ & = & E[{{\mathbb{N}}}_{2}{N}_{1}({x}_{j})]-E[{{\mathbb{N}}}_{2}]E[{N}_{1}({x}_{j})].\end{eqnarray} \tag{ 76 }$$

Here, we consider that the portion of beam 1 detected by the pixel in ${x}_{j}^{(1)}$ is locally correlated only with the corresponding portion of beam 2 at the object plane position ${x}_{j}^{(2)}$. This can be obtained, for example, if the point-to-point far-field correlations of SPDC (described in section 4.4) are imaged in beam 1 at the detection plane, while in beam 2 at the object plane. In any case, pairs of correlated spatial modes in split pseudothermal beams can be likewise used in GI experiments. Hereinafter we will omit the suffixes 1 and 2.

In the following we compare spatially incoherent, locally correlated pseudothermal beams and TWB states. The first state is usually obtained by splitting a single pseudothermal beam through a BS. In this case it holds (derived from equations (47) and (48) considering M independent modes and with the substitution ${\eta }_{j}\to {T}_{j}$):

$$\begin{eqnarray}&&\langle \hat{N}({x}_{j}){\rangle }_{{\rm{TH}}}={T}_{j}M\mu \end{eqnarray} \tag{ 77 }$$

$$\begin{eqnarray}{\langle {\delta }^{2}\hat{N}({x}_{j})\rangle }_{{\rm{TH}}} & = & {T}_{j}M\mu (1+{T}_{j}\mu )\\ & = & \langle \hat{N}({x}_{j}){\rangle }_{{\rm{TH}}}\left(1+\displaystyle \frac{\langle \hat{N}({x}_{j}){\rangle }_{{\rm{TH}}}}{M}\right)\end{eqnarray} \tag{ 78 }$$

where μ is the mean number of photons per mode, M is the number of modes detected by each pixel and T_j is the transmission coefficient in correspondence with pixel j, which can also include the detection efficiency. For $M\gg \langle \hat{N}({x}_{j})\rangle$ ($\mu \ll 1$) we have ${\langle {\delta }^{2}\hat{N}({x}_{j})\rangle }_{{\rm{TH}}}={\langle \hat{N}({x}_{j})\rangle }_{{\rm{TH}}}$: in this limit thermal light can be described by a Poisson distribution, hence approaching the shot noise.

As described in section 4, a TWB state is a quantum state of light that can be produced by the non-linear optical phenomenon of SPDC and presents perfect correlation in photon-number fluctuation. This perfect correlation is intrinsically quantum. The single beam fluctuations follow the same thermal statistics of equation (77) and equation (78).

The difference between split thermal light and TWB states arises when considering the expressions for the covariance between photon-number fluctuations in the two beams (derived from the two-modes equation (49) considering here the contribution of M pairs of independent modes):

$$\begin{eqnarray}&&\langle \delta {\hat{N}}_{2}({x}_{i})\delta {\hat{N}}_{1}({x}_{j}){\rangle }_{{\rm{TH}}}={T}_{2}({x}_{i}){T}_{1}M{\mu }^{2}{\delta }_{i,j}\end{eqnarray} \tag{ 79 }$$

$$\begin{eqnarray}&&\langle \delta {\hat{N}}_{2}({x}_{i})\delta {\hat{N}}_{1}({x}_{j}){\rangle }_{{\rm{SPDC}}}={T}_{2}({x}_{i}){T}_{1}M\mu (1+\mu ){\delta }_{i,j}\end{eqnarray} \tag{ 80 }$$

where the Kronecker delta function ${\delta }_{i,j}$ takes into account that only pairs of positions in the two beams are correlated. The transmission coefficient T₁ on channel 1 is considered uniform over the spatial resolving detector. The two statistics become asymptotically identical for $\mu \gg 1$ with a dependence $\sim {\mu }^{2}$, whereas for a small number of photons per mode ($\mu \ll 1$) TWB scales more favorably, proportional to $\sim \mu$. This means that TWB, even the shot-noise component of the fluctuations, proportional to μ, is correlated among the two beams: this is the basis of quantum enhancement.

It is now evident that, by measuring locally the covariance of each pixel with the bucket detector, it is possible to retrieve the object transmittance profile ${T}_{2}({x}_{i})$. Here, we report the demonstration for classical GI but the principle is the same for quantum GI. In fact, writing the bucket detector signal as ${{\mathbb{N}}}_{2}={\sum }_{i}{\hat{N}}_{2}({x}_{i})$ and then using equation (79), one has:

$$\begin{eqnarray}\langle S({x}_{j}){\rangle }_{{\rm{TH}}} & = & \langle \delta \hat{{{\mathbb{N}}}_{2}}\delta {\hat{N}}_{1}({x}_{j}){\rangle }_{{\rm{TH}}}\\ & = & \displaystyle \sum _{i}\langle \delta {\hat{N}}_{2}({x}_{i})\delta {\hat{N}}_{1}({x}_{j}){\rangle }_{{\rm{TH}}}\\ & = & {T}_{1}({x}_{j}){T}_{2}({x}_{j})M{\mu }^{2}={T}_{1}{T}_{2}({x}_{j})M{\mu }^{2}.\end{eqnarray} \tag{ 81 }$$

In order to quantify the quality of the reconstructed image of the object and to compare quantum and classical GI we consider the SNR. For the sake of simplicity we assume the object to be characterized by two levels of transmission, the lower one $0\leqslant {T}_{2-}\leqslant 1$, and the higher one $0\leqslant {T}_{2+}\leqslant 1$. Correspondingly, R_m is the number of pixels that are locally correlated with object points of transmission T_2m ($m=+,-$). The SNR, which represents the possibility of discerning between the two transmission coefficients, is:

$$\begin{eqnarray}&&{SNR}=\displaystyle \frac{| \langle {S}_{+}-{S}_{-}\rangle | }{\sqrt{{\delta }^{2}{S}_{+}+{\delta }^{2}{S}_{-}}}\end{eqnarray} \tag{ 82 }$$

where S_m, $m=+,-$, is the value of the correlation function (for example the covariance in equation (79)–(80)) in correspondence with T_2m. Experimentally, both the mean value, $\langle {S}_{m}\rangle$, and its variance ${\delta }^{2}{S}_{m}$ can be estimated by performing spatial averages over the regions + and − of the reconstructed image. The theoretical values of these quantities depend on the state of the light used. For evaluating $\langle {S}_{+}\rangle$ and $\langle {S}_{-}\rangle$, we use equations (79) and (80) for the classical and quantum cases, respectively. Similar to what was done in QI in equation (70) the variance of these quantities are obtained by:

$$\begin{eqnarray}\langle {\delta }^{2}{S}_{m}({x}_{j})\rangle & \equiv & \displaystyle \frac{\langle {(\delta {\hat{{\mathbb{N}}}}_{2}\delta {\hat{N}}_{1}({x}_{j}))}^{2}\rangle -\langle \delta \hat{{{\mathbb{N}}}_{2}}\delta {\hat{N}}_{1}({x}_{j}){\rangle }^{2}}{{ \mathcal K }}\\ & \simeq & \displaystyle \frac{1}{{ \mathcal K }}\langle {\delta }^{2}{\hat{{\mathbb{N}}}}_{2}\rangle \langle {\delta }^{2}{\hat{N}}_{1}({x}_{j}\in {R}_{m})\rangle .\end{eqnarray} \tag{ 83 }$$

The first equality in equation (83) is the definition of variance, while the second one holds under the hypothesis of ${R}_{m}\gg 1$; in this case the uncorrelated components dominate. Considering equation (78) and that:

$$\begin{eqnarray}\langle {\delta }^{2}{\hat{{\mathbb{N}}}}_{2}\rangle & = & \displaystyle \sum _{j}\langle {\delta }^{2}{\hat{N}}_{2}({x}_{j})\rangle \\ & = & \displaystyle \sum _{{x}_{j}\in {R}_{+}}\langle {\delta }^{2}{\hat{N}}_{2}({x}_{j})\rangle +\displaystyle \sum _{{x}_{j}\in {R}_{-}}\langle {\delta }^{2}{N}_{2}({x}_{j})\rangle \\ & = & {R}_{-}{T}_{2-}M\mu (1+{T}_{2-}\mu )+{R}_{+}{T}_{2+}M\mu (1+{T}_{2+}\mu )\end{eqnarray} \tag{ 84 }$$

we have:

$$\begin{eqnarray}\langle {\delta }^{2}{S}_{m}\rangle & = & {{ \mathcal K }}^{-1}{M}^{2}{\mu }^{2}{T}_{1}(1+{T}_{1}\mu )[{R}_{-}{T}_{2-}(1+{T}_{2-}\mu )\\ & & +{R}_{+}{T}_{2+}(1+{T}_{2+}\mu )].\end{eqnarray} \tag{ 85 }$$

It is important to note that this expression is the same both in the thermal and quantum cases; this is a consequence of the thermal nature of the single beam of a TWB state.

From the definition of SNR (equation (82)):

$$\begin{eqnarray}&&{{SNR}}_{{\rm{TH}}}\,=\,\sqrt{{ \mathcal K }}\displaystyle \frac{\sqrt{{T}_{1}}\mu ({T}_{2+}-{T}_{2-})}{\sqrt{2(1+{T}_{1}\mu )[{R}_{-}{T}_{2-}(1+{T}_{2-}\mu )+{T}_{2+}{R}_{+}(1+{T}_{2+}\mu )]}}\end{eqnarray} \tag{ 86 }$$

$$\begin{eqnarray}&&{{SNR}}_{{\rm{SPDC}}}\,=\,\sqrt{{ \mathcal K }}\displaystyle \frac{\sqrt{{T}_{1}}(1+\mu )({T}_{2+}-{T}_{2-})}{\sqrt{2(1+{T}_{1}\mu )[{R}_{-}{T}_{2-}(1+{T}_{2-}\mu )+{T}_{2+}{R}_{+}(1+{T}_{2+}\mu )]}}.\end{eqnarray} \tag{ 87 }$$

As expected the SNR increases with the number of acquisitions ${ \mathcal K }$ (namely the total number of frames collected by the spatial resolving detector), while the dependence on other parameters is more complex. For a better understanding of these expressions, the relevant physical limits are analyzed.

• Let us consider the case of $\mu \gg 1$, which is the situation of a high number of photons per mode. In order to further simplify the expressions we also consider the case of ${T}_{2+}=1$ and ${T}_{2-}=0$ (the detector detects all the photons that do not hit the object, modeled as perfectly absorbing mask):
  $$\begin{eqnarray}\mu \gg 1:{{SNR}}_{{\rm{TH}}} & = & {{SNR}}_{{\rm{SPDC}}}\\ & = & \sqrt{{ \mathcal K }}\displaystyle \frac{{T}_{2+}-{T}_{2-}}{\sqrt{2}\sqrt{{T}_{2+}^{2}{R}_{+}+{T}_{2-}^{2}{R}_{-}}}\end{eqnarray} \tag{ 88 }$$
  $$\begin{eqnarray}{T}_{2+}=1;{T}_{2-} & = & 0:{{SNR}}_{{\rm{TH}}}\\ & = & {{SNR}}_{{\rm{SPDC}}}=\displaystyle \frac{\sqrt{{ \mathcal K }}}{\sqrt{2{R}_{+}}}.\end{eqnarray} \tag{ 89 }$$
  In this limit the same expressions are found both in the classical or quantum case. This result is not surprising and comes from the fact that for $\mu \gg 1$ the expressions for the covariances converge asymptotically. Looking at equation (88) it results that in this limit the SNR does not depend on the transmittance on channel ${\text{}}1$, T₁.
• In the opposite case, for $\mu \ll 1$, the expressions become:
  $$\begin{eqnarray}&&\mu \ll 1:{{SNR}}_{{\rm{TH}}}=\sqrt{{ \mathcal K }}\displaystyle \frac{\sqrt{{T}_{1}}({T}_{2+}-{T}_{2-})}{\sqrt{2[{R}_{-}{T}_{2-}+{T}_{2+}{R}_{+}]}}\mu \end{eqnarray} \tag{ 90 }$$
  $$\begin{eqnarray}&&{{SNR}}_{{\rm{SPDC}}}=\sqrt{{ \mathcal K }}\displaystyle \frac{\sqrt{{T}_{1}}({T}_{2+}-{T}_{2-})}{\sqrt{2[{R}_{-}{T}_{2-}+{T}_{2+}{R}_{+}]}}\end{eqnarray} \tag{ 91 }$$
  $$\begin{eqnarray}&&\mathrm{for}\,{T}_{2+}=1\,\mathrm{and}\,{T}_{2-}=0:{{SNR}}_{{\rm{SPDC}}}=\displaystyle \frac{\sqrt{{ \mathcal K }}\sqrt{{T}_{1}}}{\sqrt{2{R}_{+}}}.\end{eqnarray} \tag{ 92 }$$
  In this limit the difference between the two cases is evident: while in the classical case the SNR decreases proportionally with μ and approaches 0 as soon as $\mu \longrightarrow 0$, in the quantum case the SNR converges to a constant value. It is important to notice that this constant depends on ${ \mathcal K }$, and can therefore be arbitrary increased with a longer acquisition time experiment.

Moreover, in both regimes considering ${T}_{2+}=1$ and ${T}_{2-}=0$, ${SNR}\propto \tfrac{1}{\sqrt{{R}_{+}}}$. For a fixed total area, a greater ${R}_{+}$ implies a smaller ${R}_{-}$, hence a small object; this result therefore explains the reasonable fact that it is more difficult to image a small object. We also recall that all these expressions for the SNR are obtained in the limit ${R}_{+}\gg 1$.

To conclude the comparison between classical and quantum GI we consider the ratio, G, of the two SNRs. From the exact expressions in equations (86) and (87), it follows that:

$$\begin{eqnarray}&&G=\displaystyle \frac{{{SNR}}_{{\rm{SPDC}}}}{{{SNR}}_{{\rm{TH}}}}=\displaystyle \frac{1}{\mu }+1.\end{eqnarray} \tag{ 93 }$$

• For $\mu \gg 1$, $G\longrightarrow 1$: in this regime, the quantum and classical case are equivalent in terms of SNR. The quantum enhancement is in this case negligible.
• For $\mu \ll 1$, $G\longrightarrow \infty$: this is the regime where the quantum enhancement is more important. Only using TWB states it is possible to retrieve the object profile.

As we pointed out in the introduction the analogy between GI and QI is confirmed by equation (93), which is exactly the same as the one in equation (73).

Sometimes, in practical situations it could be helpful to compare the SNR for quantum and classical GI, as a function of the detected photons instead of the photon per mode emitted by the source. For this purpose we consider for example the detected photons per pixel of the spatially resolved detector $\langle {\hat{N}}_{1}\rangle ={T}_{1}M\mu$. This can be the quantity of interest if there is a strong limitation on the total photon that can be used per acquisition time, for example in the case of a detector with a low level of saturation, or to not exceed some damage-level of a photosensitive sample. In terms of $\langle {\hat{N}}_{1}\rangle$ equation (93) becomes:

$$\begin{eqnarray}&&G=\displaystyle \frac{{{SNR}}_{{\rm{SPDC}}}}{{{SNR}}_{{\rm{TH}}}}=1+\displaystyle \frac{{T}_{1}M}{\langle {\hat{N}}_{1}\rangle }.\end{eqnarray} \tag{ 94 }$$

A clear advantage of the quantum GI appears when less than one photon per pixel is detected (for ideal efficiency, ${T}_{1}=1$). However, the higher the number of spatio-temporal modes M collected by each pixel, the higher the quantum enhancement effect.

Here, we have presented the derivation of the SNR for the specific case in which the covariance of the bucket and the spatial resolving detector is used for image reconstruction. Different correlation functions can be used and can be more advantageous in a particular case, for example in the case of a slightly absorbing object [172]. The case treated here is sufficient for the purpose of this review, which is to identify the regimes in which quantum light is advantageous and quantifying this effect.

This consideration paves the way for a lot of interesting applications of quantum GI in situations where a low light level is needed, like in the case of imaging of certain biological samples.

Finally, we note that similarities between GI and QI performed using intensity measurements (described in section 6.1) arise. Comparing figures 10 and 14 the analogy is evident. In particular GI can be seen as a specific case of QI intensity protocol, where the background at the bucket detector comes from those spatial modes of the source which are not correlated with the single pixel of the spatial resolving detector. Of course, in GI the spatial resolution of the reference arm allows a full reconstruction of the object transmission profile, while the QI goal is just discriminating its presence. Despite this difference, the quantum enhancement in terms of SNR assumes the same form.

The first calibration method exploiting quantum correlations was proposed in the seventies by Klyshko [98, 186], but only in the nineties did the technological development allow accurate calibrations to be performed [187]. Nowadays, Klyshko's calibration technique is recognized as a fundamental metrological tool by the international radiometric community [60, 188–194].

The Klyshko method is based on the SPDC process described in section 4. Photons emitted by SPDC are always produced in pairs almost simultaneously, the presence of n photons, in a particular set of optical modes, guarantees the presence of exactly n photons in the set of conjugated optical modes. We label the two sets of conjugated optical modes, and the corresponding paths, as 1 and 2.

In the basilar Klyshko technique two click/no-click single-photon detectors are used, as shown in figure 15. The detector in path 1 is the device under test (DUT) while the second, on path 2, is the reference. In a real measurement, the clicks on the reference channel are used to trigger a coincidence circuit. If we assume that the losses in the system are only due to the non-ideal quantum efficiency of the detectors and that dark counts and the background photon level are negligible, then the calibration technique is quite simple: by definition the number of photons detected by the DUT and by the trigger detector are, respectively, ${n}_{1}={\eta }_{1}n$ and ${n}_{2}={\eta }_{2}n$, and the number of times both detectors click (coincidences) is simply $C={\eta }_{1}{\eta }_{2}n$. It is possible to determine the efficiency ${\eta }_{1}$ simply by taking the ratio ${\eta }_{1}=C/{n}_{2}$.

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When real devices are considered, this simple formula has to be modified to account for the presence of noise and losses. The number of measured counts should be corrected by the sum n_B of the electronic dark counts and spurious photo-counts that are not correlated among the paths. The true coincidences have to be obtained from the measured ones C_m by subtracting the accidental coincidences C_A occurring between two dark counts, a dark count and a photon, or between two uncorrelated photons. Concerning the losses, it is important to note that the methods cannot distinguish between losses due to the optical elements along the path, parametrized here by τ ($0\leqslant \tau \leqslant 1$), and the ones occurring at the DUT, which represent the quantum efficiency ${\eta }_{1}$. In any case, an independent calibration of the transmissivity of all the optical elements allows an independent estimation of the path transmissivity τ to be performed.

Combining losses and noise effects, the DUT detector efficiency can be estimated as:

$$\begin{eqnarray}&&{\eta }_{1}=\displaystyle \frac{{C}_{{\rm{m}}}-{C}_{{\rm{A}}}}{\tau ({n}_{2}-{n}_{{\rm{B}}})},\end{eqnarray} \tag{ 95 }$$

in which all the quantities are directly measurable. In particular, we note that the efficiency of the trigger detector and the number of photon pairs generated do not appear in the final equation, confirming that Klyshko's calibration is an absolute technique in which no pre-calibrated references are needed. Finally, up to now we have not pointed out issues arising from the saturation of click/no-click detectors and from the typical inactivity time (dead time) of these detectors after a counting event. For an accurate calibration of these devices it is necessary to take into account, and correct for, the dead time and operate the source at very low intensity so that the probability of emitting more than one pair of photons in the detection window is negligible.

Recently, many variants and extensions of the Klyshko technique for different intensity regimes and other type of detectors have been studied. The possibility of exploiting the quantum correlations at higher photon fluxes for calibrating analog and linear devices [195–198] have been investigated and application to spatial resolving detectors, such as multi-pixel cameras, has been demonstrated with good accuracy. We will discuss these advancements in the next sections.

PNR detectors play an important role in many fields of science and technology [199, 200]. PNR capabilities can be built upon multiplexing click/no-click detectors [201–205] or using intrinsically PNR detectors such as photo-multipliers [206–209], visible light photon counters [210, 211], transition edge sensors (TESs) [212] and inductive superconducting transition edge detectors [213].

The basic version of the Klyshko technique can also be used to calibrate PNR detectors. However, the direct application of such a method does not exploit the full potential of a PNR detector because it does not take into account the possibility of having more photons simultaneously. An extension of the Klyshko method that involves contribution of more than one photon pair at a time has been developed recently [214]. The apparatus, shown in figure 16, is similar to the apparatus of the basic Klyshko technique. The main differences are: the DUT is a TES, i.e. a superconducting PNR detector, and the intensity of SPDC emission is sufficient to produce more then one pair of photons in the detection window. The detector used as a trigger is still a click/no-click detector with unknown quantum efficiency. The typical histogram representing the output of a PNR detector in terms of relative frequency of detection events as a function of the electric pulse amplitude (corresponding to the number of photons) is shown in figure 17.

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To perform an absolute calibration it is necessary to acquire two separate sets of measures, one in the presence and one in the absence of heralding photons (i.e. photons detected by the trigger detector). It is then possible to estimate the probabilities of observing i counts in the presence and in the absence of the heralded photon, indicated respectively by P(i) and ${ \mathcal P }(i)$.

Now it is useful to define the following quantities: the probability of having a true heralding count ξ (i.e. not due to noise), the overall quantum efficiency γ (including detector efficiency and channel losses), the transmissivity of the optical channel τ from the crystal to the PNR detector, and the quantum efficiency of the detector itself η. From these definitions we can write the relation: $\gamma =\tau \eta$.

The PNR detector has a probability of observing no photons and i photons given, respectively, by:

$$\begin{eqnarray}&&P(0)=\xi [(1-\gamma ){ \mathcal P }(0)]+(1-\xi ){ \mathcal P }(0)\end{eqnarray} \tag{ 96 }$$

$$\begin{eqnarray}&&P(i)=\xi [(1-\gamma ){ \mathcal P }(i)+\gamma { \mathcal P }(i-1)]+(1-\xi ){ \mathcal P }(i).\end{eqnarray} \tag{ 97 }$$

Inverting these two equations it is possible derive the overall quantum efficiency of the PNR detector, which includes the losses on the path. Note that, different to the original Klyshko technique, there are several ways of calculating the quantum efficiency: one for each peak observed in the histogram:

$$\begin{eqnarray}&&{\gamma }_{0}=\displaystyle \frac{{ \mathcal P }(0)-P(0)}{\xi { \mathcal P }(0)},\qquad {\gamma }_{i}=\displaystyle \frac{P(i)-{ \mathcal P }(i)}{\xi [{ \mathcal P }(i-1)-{ \mathcal P }(i)]}.\end{eqnarray} \tag{ 98 }$$

Each of these derivations of the quantum efficiency exploits a different number of simultaneously detected photons and can be calculated independently. However, all the ${\gamma }_{i}$ represent the same physical quantity and should have the same value for a linear detector. Therefore, this extension of the Klyshko technique allows the consistency of the detection model with data comparing different ${\gamma }_{i}$ to be checked. In addition, in this case, to obtain the quantum efficiency of the detector, $\eta =\gamma /\tau$, it is necessary to estimate independently the losses on the optical path.

The techniques described in sections 8.1 and 8.2 are based on true single-photon and PNR detectors, which allow us to measure temporal coincidences between the time tagged photo-counting events. However, these devices cannot be used for detecting beams with relatively high photon flux, they are expensive and require complex coincidence electronics. On the other hand, the majority of optical detectors operate in the analog regime, providing an output signal that is proportional to the intensity of the light, namely to the number of photons, and have a large dynamic range. In contrast, the high electronic background noise does not allow the individual photons to be discriminated and coincidence measurements to be performed. This limitation does not prevent the possibility of observing true quantum effects such as SSN intensity correlations, when the shot-noise fluctuations proportional to $\sqrt{N}$ emerge from the background noise. On the one hand, this enables the use of analog detectors for quantum enhanced imaging and sensing applications (see for example sections 5, 6, and 7) and on the other hand it allows the absolute calibration of the devices [120, 195–198, 215].

The calibration method discussed here is strongly based on the detection of a set of pair-wise correlated spatio-temporal modes of twin beams impinging the detectors areas, according to the model described in section 4.4. There, we demonstrated the relation between the measured noise reduction factor ${\sigma }_{{\rm{\det }}}$ and the detection efficiencies in the case where they are perfectly balanced, see equation (56). Here we report the more general formula in which the detectors have different efficiencies ${\eta }_{1}$ and ${\eta }_{2}$ [90, 120]:

$$\begin{eqnarray}&&{\sigma }_{\alpha }\simeq \displaystyle \frac{1+\alpha }{2}-{\eta }_{1}A\end{eqnarray} \tag{ 99 }$$

where $\alpha =\langle {\hat{n}}_{1}\rangle /\langle {\hat{n}}_{2}\rangle ={\eta }_{1}/{\eta }_{2}$ is the measurable ratio between the beams intensities and A is the collection efficiency introduced in equations (57) and (61), a function of geometrical parameters (essentially the size of the detector area and the coherence area) that can be measured independently. Moreover, in equation (99) we introduced a slightly modified NRF parameter ${\sigma }_{\alpha }$, to compensate unbalancing [216]:

$$\begin{eqnarray}&&{\sigma }_{\alpha }=\displaystyle \frac{\langle \delta {({\hat{n}}_{1}-\alpha {\hat{n}}_{2})}^{2}\rangle }{\langle {\hat{n}}_{1}+\alpha {\hat{n}}_{2}\rangle }.\end{eqnarray} \tag{ 100 }$$

The absolute value of the efficiency ${\eta }_{1}$ obtained by inverting equation (99) is a mean value over the whole area of detection. Note that in the first approximation this equation is valid whatever photon flux impinges the detectors and even for high-gain SPDC, where many photon pairs are generated in a single spatio-temporal mode. A more accurate analysis discloses that there is a certain dependence of the collection efficiency A from the mean occupation number μ of the uncorrelated modes ${{ \mathcal M }}_{{\rm{u}}}$, as reported in equation (57), but this dependence can be minimized as long as a careful matching of the conjugated mode between the detectors is performed.

The technique described above is particularly suitable for calibrating an analog (but not only analog, as we will see in the next section) spatial resolving detector, such as CCD cameras. One of the reasons is the possibility of setting arbitrarily the detection areas and to have fine control in the collection of the conjugated spatial modes. Nevertheless, calibration techniques for spatial resolving detectors are essential for many applications, among which imaging represents the most significant. Due to their importance, in the following we focus our attention on CCD cameras, but, in principle, the technique can be applied to any spatial resolving detector that provides, point by point, an analogical signal proportional to the impinging light flux. Standard CCD cameras, i.e. without any avalanche electro-multiplication, are able to count the number of photoelectrons generated in each pixel for a given exposure time, providing an output proportional to the intensity of the adsorbed light (analogical regime). There are two main sources of noise in CCD cameras: thermal noise and read noise. Thermal noise is due to charges generated by thermal excitations, which is proportional to the exposure time and it is strongly dependent on the temperature. Read noise is generated in the electronic reading process and it is independent of the exposure time and other physical parameters. The read noise contribution is not avoidable and its presence implies that CCDs cannot distinguish single photons from the background. In 2010, the first absolute calibration of a standard CCD camera was realized by exploiting a bright squeezed vacuum [90] and, after several improvements, this technique reached a level of accuracy suitable for metrological application [216], aligned with the state of the art of the absolute calibration of single-photon detectors using Klyshko's method.

An EMCCD is a camera able to detect single photons with high quantum efficiency. This capability is achieved exploiting an electron multiplication structure, built into the sensor, that can be activated or not, giving the possibility to switch from the analog to the single-photon regime using the same device.

The statistical distribution of the output counts for this device is well understood [217, 218]. Given n photoelectrons generated inside a pixel, the multiplication stage provides a random number of electron counts x following the distribution:

$$\begin{eqnarray}&&{ \mathcal P }(x| n)=\displaystyle \frac{{x}^{n-1}\exp (-x/g)}{{g}^{n}(n-1)!}\ \ \ \ \ {\rm{for}}\ \ \ n\gt 0;\end{eqnarray} \tag{ 101 }$$

$$\begin{eqnarray}&&{ \mathcal P }(x| n)=\delta (x)\ \ \ \ \ \mathrm{for}\ \ \ n=0;\end{eqnarray} \tag{ 102 }$$

where g is the multiplication gain. The total number of counts per pixel is due to the contribution of photoelectrons and noise. Therefore, the count distribution at the output is the convolution of ${ \mathcal P }(x| n)$ with the noise distribution:

$$\begin{eqnarray}&&{P}_{{\rm{tot}}}(x| n)={\int }_{0}^{\infty }{ \mathcal P }(y| n){P}_{{\rm{noise}}}(x-y){\rm{d}}{y}.\end{eqnarray} \tag{ 103 }$$

The statistical distribution of the output counts implies that, despite the possibility of detecting single photons, each pixel of an EMCCD is not a native photon-counting detector. It is possible to use an EMCCD in a photon counting regime by means of proper data processing, after which each pixel can be considered as a click/no-click detector. This behaviour is achieved applying a discriminating threshold T on the electron counts x at each pixel: a photon is detected when $x\gt T$. In this regime, a detection area ${{ \mathcal A }}_{{\rm{\det }}}$ on the sensor can be used as a non-linear photon-number-resolving detector, counting the number of pixels that have $x\gt T$ (spatial multiplexing).

As demonstrated in a recent experimental work [61], exploiting the spatially multimode quantum correlations in squeezed vacuum states, it is possible to calibrate an EMCCD, both in an analog regime and in a photon counting regime, obtaining two different quantum efficiencies, indicated respectively by ${\eta }_{0}$ and $\eta (T)$. The quantum efficiency in a single-photon counting regime is strongly dependent on the threshold T in a predictable way. Moreover, it has been demonstrated that there exists a relation between the quantum efficiencies ${\eta }_{0}$ and $\eta (T)$, by providing a radiometric link between the low illumination range to the mesoscopic and to the macroscopic range [61]. This result represents an important step in the field of quantum radiometry, in particular because it allows the metrological traceability of measurements at the few-photon level, which is essential for most emerging quantum technologies.

The calibration of the analog quantum efficiency ${\eta }_{0}$ is identical for EMCCDs and for standard CCDs, and was described in section 8.3. In addition, the calibration method for EMCCDs in the photon-counting regime is based on the same principle of calibration as in the analog regime, therefore the experimental apparatus is the same as in figure 18. Indeed, EMCCD calibration is also based on the measurement of the corrected NRF reported in equation (100). However, we have to take into account that, in this case, n₁ and n₂ are the numbers of pixels that have $x\gt T$ in two correlated areas and they depend on the threshold:

$$\begin{eqnarray}&&{\sigma }_{\alpha }(T)=\displaystyle \frac{\langle \delta {({\hat{n}}_{1}(T)-\alpha {\hat{n}}_{2}(T))}^{2}\rangle }{\langle {\hat{n}}_{1}(T)+\alpha {\hat{n}}_{i}(T)\rangle }.\end{eqnarray} \tag{ 104 }$$

This quantity satisfies the relation with the quantum efficiency as reported in equation (99):

$$\begin{eqnarray}&&{\sigma }_{\alpha }(T)\simeq \displaystyle \frac{1+\alpha }{2}-\eta (T)A.\end{eqnarray} \tag{ 105 }$$

Therefore, it is possible to use the same absolute calibration technique, both for the photon-counting regime and the analogical regime.

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In principle, for an EMCCD operating in a photon-counting regime, it is also possible to exploit directly the Klyshko method (as in section 8.1) to perform an absolute calibration of the quantum efficiency. However, there are two main practical reasons that prevent its use. First of all, Klyshko's technique needs a few coincidences per frame and unfortunately, in this configuration, the noise in EMCCDs becomes dominant, preventing any possible coincidences being counted. The second reason is that the read time of an EMCCD is much higher with respect to typical single-photon detectors, and as a consequence the Klyshko technique would be much too slow for practical applications.

In this section we focused our attention on the most diffuse spatial resolving devices that allow single-photon counting: EMCCD cameras. However, other types of spatial resolving detectors are able to work in the photon-counting regime. An important commercial device is the ICCD, for which similar absolute calibration techniques have been developed [62, 63]. Most recently, a spatial resolving detector based on arrays of true 'click/no click' single photon detectors has been developed [219]. This type of device, largely used in recent quantum optics experiments [220, 221], can directly exploit the calibration techniques based on squeezed vacuum correlations.