1. Introduction

In the last 40 years, fields composed of entangled photon pairs and generated in spontaneous parametric downconversion [1] have represented a workhorse in the area of experimental quantum optics devoted to demonstrations of quantum properties of nature as well as realizations of various applications/protocols exploiting these properties in quantum secure communications [2], ultra-precise metrology [3], and quantum computation [4,5]. Whereas tests of the Bell inequalities [6], quantum teleportation [7] of a photon state, and dense coding [8] belong to the former, quantum-key-distribution systems [2,9], method of absolute detector calibration [10], sub-shot-noise metrology [11], and realizations of various kinds of quantum gates [12] can be mentioned as useful examples of the latter. Exploitation of photon-pair entanglement [1] is their common feature.

On the other hand, more intense fields composed of several or even more photon pairs are available in the same nonlinear process when pumped more intensively. These fields, that are called twin beams (TWBs) (for the review, see [13]), are also endowed with ideally perfect photon-number cross-correlations in their constituting signal and idler beams [1]. This property, though seemingly very classical, in fact guarantees their highly nonclassical properties. This can be evidenced from the form of the corresponding integrated-intensity quasi-distributions [14,15] with negative probability densities and fast oscillations [16]. Experimental verification of these sub-shot-noise photon-number correlations was reached in Refs. [17–20]. These quantum correlations have been exploited for absolute detector calibration [21], in virtual-state entangled-photon spectroscopy [22–25], or in quantum-key-distribution systems [26]. The use of such states in the virtual-state entangled-photon spectroscopy reveals a certain limitation stemming from their properties: their marginal photon-number distributions are multi-mode thermal, i.e., they exhibit broad (classical) photon-number fluctuations in the individual beams. This poses a question about the construction and generation of a new kind of fields with even better properties—sub-Poissonian twin beams with marginal photon-number fluctuations reduced below the shot-noise level.

Such states can indeed be reached using the method of photon addition [27,28]. This method was originally introduced in Ref. [27] in the area of cavity quantum electrodynamics (QED) [29] where the added photons were provided by excited atoms. Later a beam splitter was suggested for photon addition [30]. However, the first experimental realization exploited optical parametric amplification [31,32]. Among others and using photon-added thermal states, quantum commutation relations [33,34] were experimentally confirmed. Photon addition has already been found useful in enhancing the performance of quantum teleportation [35,36], entanglement distillation [37], testing non-Gaussianity [38], and entanglement [39], ultra-precise measurement of the phase [40], and generation of the Schrödinger cat states [41] as well as macroscopic entangled states [42]. More details about photon addition can be obtained in Refs. [43,44].

We note that there also exists the scheme for photon subtraction [45] using a beam splitter. When photons are subtracted from both beams of a TWB, sub-Poissonianity in the individual beams can be reached [46], similarly as when photon addition is applied. However, in this case, tight photon-number correlations in the TWB are necessary to arrive at the marginal sub-Poissonian photon-number distributions. Multi-mode TWBs are needed for practical experimental realization [47]. Though the theory predicts the possibility of having marginal sub-Poissonian photon-number distributions even when a TWB is composed just of two modes, i.e., it is in a squeezed two-mode vacuum state, the experimental realization seems to be too demanding [48].

2. Sub-Poissonian Twin Beams: Simple Model and Feasible Experimental Scheme

We demonstrate photon addition as a tool for reaching sub-Poissonian TWBs by considering an ideally paired multi-mode TWB. The state of such TWB is in general mixed and it is conveniently described by its joint signal-idler photon-number distribution $p^{\textrm{TWB}}_{\textrm{si}}(n_{\textrm{s}},n_{\textrm{i}})$ in its ideal form $p^{\textrm{TWB}}_{\textrm{id}}(n_{\textrm{s}}) \delta _{n_{\textrm{s}},n_{\textrm{i}}}$, where $\delta$ denotes the Kronecker symbol and $n_{\textrm{s}}$ ($n_{\textrm{i}}$) stands for the signal (idler) photon number. Addition of $n$ photons into both beams in the TWB then leads to the following photon-number distribution: $p^{\textrm{add}}_{\textrm{si}}(n_{\textrm{s}},n_{\textrm{i}};n,n) = p^{\textrm{TWB}}_{\textrm{id}}(n_{\textrm{s}}-n) \delta _{n_{\textrm{s}},n_{\textrm{i}}}$ for $n_{\textrm{s}},n_{\textrm{i}} \ge n$ and being zero otherwise. Whereas the noise-reduction parameter $R_n$,

For experimental implementation, either two-mode squeezed vacuum states [49] or multi-mode TWBs [13] may be considered as the original states ready for photon addition. Whereas a photon has to be added into a two-mode squeezed vacuum state coherently using an interferometric setup [28,48], photon addition into a multi-mode TWB can be accomplished incoherently. This brings a significant simplification of the experimental setup, especially when the photons to be added originate in other TWBs to which post-selection [50] conditioned by detecting a given number of photons in one of their beams is applied [for the scheme, see Fig. 1(a)] [51–53]. We note that incoherent photon addition may weaken some of the quantum properties of the obtained states, similarly as in the case of compound TWBs studied in Ref. [54]. In the experiment, declination from the above ideal model occurs owing to the limited detection efficiencies of the real post-selecting photon-number-resolving detectors: sub-Poissonian fields [1] are added into the original TWB instead of the fields in the Fock states with given photon numbers. The photon-added twin beams (PATWBs) obtained after detecting $\bar {c}_{\textrm{s}}$ [$\bar {c}_{\textrm{i}}$] signal [idler] photocounts when preparing the state added into the signal [idler] beam are then characterized by their joint photon-number distributions $p_{\textrm{si}} (n_{\textrm{s}},n_{\textrm{i}};\bar {c}_{\textrm{s}},\bar {c}_{\textrm{i}})$. These joint photon-number distributions are monitored in the experiment by photon-number resolving detectors that provide us the joint photocount histograms $f_{\textrm{si}}(c_{\textrm{s}},c_{\textrm{i}};\bar {c}_{\textrm{s}},\bar {c}_{\textrm{i}})$ of PATWBs determined along the formula

 

Fig. 1. (a) Scheme for generating sub-Poissonian twin beams: three nonlinear crystals NLC${}_{\rm p,s,i}$ generate twin beams TWB${}_{\rm p,s,i}$. Using photon-number-resolving detector $\bar {\textrm{D}}_{\textrm{s}}$ ($\bar {\textrm{D}}_{\textrm{i}}$) in the idler (signal) beam of twin beam TWB${}_{\textrm{s}}$ (TWB${}_{\textrm{i}}$) a potentially sub-Poissonian state is reached in the signal (idler) beam and then it is combined with the signal [idler] beam of twin beam TWB${}_{\textrm{p}}$ to build a PATWB. Detection efficiencies $\eta$, numbers $N$ of pixels in detection areas and mean dark counts $d$ per pixel characterize the used detectors. (b) Experimental setup: a nonlinear crystal BBO is used to generate TWBs by ultra-short third-harmonic pulses (THG) with actively stabilized intensity using rotating half-wave plate (HWP), polarizing beam splitter (PBS), and detector (DET) as an active feedback. The idler beam is reflected off a highly reflecting mirror (HR), both beams are spectrally filtered using bandpass interference filter and detected by an iCCD camera. (c) Four detection areas are defined on the photocathode of the iCCD camera with multiply exposed signal and idler strips: ${\textrm{D}}_{\textrm{s}}$ and ${\textrm{D}}_{\textrm{i}}$ serve for monitoring PATWBs, $\bar {\textrm{D}}_{\textrm{s}}$ ($\bar {\textrm{D}}_{\textrm{i}}$) is used for post-selecting the field added into the signal (idler) beam. Arrows indicate photon pairing as well as tight spatial correlations in the corresponding detection areas.

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Detection efficiencies $\bar {\eta }_{\textrm{s}}$ and $\bar {\eta }_{\textrm{i}}$ of the post-selecting detectors used in the preparation of sub-Poissonian states are critical for reaching highly nonclassical and highly entangled PATWBs. The reason is that the decreasing detection efficiency $\bar {\eta }$ broadens the photon-number fluctuations which results in the simultaneous increase of the Fano factor $F_n$ and noise-reduction parameter $R_n$ of PATWBs, as quantified in the graphs of Fig. 2 considering three noiseless TWBs each having 2.5 mean numbers of photon pairs equally distributed into 200 independent modes. The curves in Fig. 2 predict that the detectors with $\bar {\eta } \approx 0.2$ allow for the generation of PATWBs with $F_n \approx 0.6$ and $R_n \approx 0.3$ when triggered by the realistic post-selecting photocount numbers $\bar {c} \approx 3$.

 

Fig. 2. (a) Fano factor $F_{n,{\textrm{s}}}$ and (b) noise-reduction parameter $R_n$ as they depend on detection efficiency $\bar {\eta } \equiv \bar {\eta }_{\textrm{s}} = \bar {\eta }_{\textrm{i}}$ and number $\bar {c}\equiv \bar {c}_{\textrm{s}}=\bar {c}_{\textrm{i}}$ of photocounts conditioning the generation of PATWBs. Three noiseless TWBs with $B^{\textrm{p}}_a = 0.0125$ and $M^{\textrm{p}}_a = 200$, $a = {\rm p,s,i}$, and detectors without dark counts are assumed (for details, see Supplement 1).

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Moreover, the model with its graphs plotted in Fig. 2 suggests $F_n \approx 0.5$ and $R_n \approx 0.01$ for $\bar {\eta } \approx 1$ which can be experimentally simulated once the tight spatial correlations in the signal- and idler-beam photocounts are recorded by the iCCD camera. Provided that the mean photon-pair number per one spatial mode of a TWB is much lower than one, tight spatial correlations of photons in a pair inside a TWB allow us to identify pairs of photocounts emerging after detecting both photons from one photon pair. If photocounts that are not paired this way are excluded from the experimental data, the remaining paired photocounts ideally correspond to the measurement with detection efficiency $\bar {\eta } \approx 1$. In reality, the intensity cross-correlation functions of TWBs that quantify the spatial correlations of the signal and the idler photons in a common photon pair are blurred and parameters for identifying such photocount pairs have to be suitably chosen (for details, see Ref. [57]). Then, TWBs with nearly ideal photon-number correlations can be obtained [58].

The use of an iCCD camera as a photon-number-resolving detector is a striking feature of our setup for two reasons. (i) It allows for a significant simplification of the experimental geometry as all four detectors found in the principal scheme in Fig. 1(a) are realized in different detection areas at the photocathode of one iCCD camera. This is shown in Figs. 1(b) and1(c) depicting the experimental setup and detailed partitioning of the camera photocathode. (ii) An iCCD camera registering photocounts in the transverse beams plane allows for monitoring the spatial pairing of photocounts in the twin beams TWB${}_{\textrm{s}}$ and TWB${}_{\textrm{i}}$ exploited in the post-selection. This allows us to eliminate in the post-selection the cases in which both photons from a photon pair were not detected and simulate this way the operation of a nearly ideal photon-number-resolving detector.

PATWBs were generated in the experiment whose setup is shown in Fig. 1(b). An iCCD camera Andor DH734-18F-63 used as a multichannel photon-number-resolving detector replaces four detectors ${\textrm{D}}_{\textrm{s}}$, ${\textrm{D}}_{\textrm{i}}$, $\bar {\textrm{D}}_{\textrm{s}}$, and $\bar {\textrm{D}}_{\textrm{i}}$ from the principal scheme shown in Fig. 1(a). This results in considerable simplification of the setup. The iCCD camera set for the gating time of 4 ns (given by the electron multichannel intensifier) was triggered by electronic pulses coming from the laser. Three TWBs from the setup in Fig. 1(a) were realized at different neighboring parts of the signal- and idler-beam emission cones of type-I spontaneous parametric downconversion in a 5-mm-long beta-barium-borate crystal (BaB${}_2$O${}_4$, BBO) cut for a slightly non-collinear geometry. The nonlinear process was pumped by the third-harmonic pulses (280 nm) initiated by the pulses of a femtosecond cavity-dumped Ti:sapphire laser (840 nm, 150 fs). The signal- and idler-beam emission cones were constrained by a 14-nm-wide bandpass interference filter in front of the iCCD camera that spectrally filters the beams at nearly frequency-degenerated frequencies $\approx 560$ nm. The signal and the idler beams as observed at the photocathode were divided into three equally illuminated parts that correspond to three TWBs used in the scheme for generating PATWBs [see partitioning of the photocathode in Fig. 1(c)]. Whereas one detection area in the signal [idler] beam played the role of post-selecting detector $\bar {\textrm{D}}_{\textrm{i}}$ [$\bar {\textrm{D}}_{\textrm{s}}$], two remaining detection areas in this beam merged together to form detector ${\textrm{D}}_{\textrm{s}}$ [${\textrm{D}}_{\textrm{i}}$]. Parameters characterizing these four detection areas were determined using the method of detector calibration by TWBs [21] and independent measurements. The detection efficiencies of post-selecting detectors and those measuring the photocount distributions of PATWBs were identified as $\eta _{\textrm{s}}=\bar {\eta _{\textrm{i}}} = 0.234\pm 0.005$ and $\eta _{\textrm{i}}=\bar {\eta _{\textrm{s}}} = 0.227\pm 0.005$. The calibration method assumes a TWB in a general multi-mode Gaussian form [59] composed of three components describing (i) ideal photon pairs, (ii) single photons in the signal beam, and (iii) single photons in the idler beam. Each of these components is modelled as a multi-mode thermal field with a given number of equally populated modes. Determining all these parameters, the calibration method provided us the mean numbers of photon pairs of the used TWBs: $\langle n_{\textrm{p}}^{\textrm{p}}\rangle = 2.63\pm 0.05$; $\langle n^{\textrm{p}}_{\textrm{s}}\rangle = 2.25\pm 0.05$; and $\langle n^{\textrm{p}}_{\textrm{i}}\rangle = 2.28\pm 0.05$. Mean numbers of single signal and single idler photons were less than 0.4 in all TWBs, i.e., they were much lower than the mean numbers of photon pairs. More details are given in the Supplement 1 and Table 1.

Table 1. Probability $p^{\textrm{p}}$ (in %) of Generating a PATWB after Detecting $\bar{c}_{\rm s}$,$\bar{c}_{\rm i}$ Photocounts at the Post-selecting Detectors $\bar {\textrm{D}}_{\textrm{s}}$ and $\bar {\textrm{D}}_{\textrm{i}}$, Respectively^a

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Distortions caused by the detection efficiencies $\eta _{\textrm{s}}$ and $\eta _{\textrm{i}}$ in the experimental PATWB histograms $f_{\textrm{si}}(c_{\textrm{s}},c_{\textrm{i}};\bar {c}_{\textrm{s}},\bar {c}_{\textrm{i}})$ were then eliminated by the maximum-likelihood reconstruction [55,60] of the corresponding photon-number distributions $p_{\textrm{si}}(n_{\textrm{s}},n_{\textrm{i}};\bar {c}_{\textrm{s}},\bar {c}_{\textrm{i}})$. Even the detrimental effects of the limited detection efficiencies $\bar {\eta }_{\textrm{s}}$ and $\bar {\eta }_{\textrm{i}}$ of the post-selecting detectors were suppressed to high extent once the tight spatial correlations of photons in the twin beams TWB${}_{\textrm{s}}$ and TWB${}_{\textrm{i}}$ were exploited [57]. This was achieved once we counted for post-selection only the photocounts that had the spatially correlated photocounts in the corresponding detection area ($\bar {\textrm{D}}_{\textrm{s}}$ in ${\textrm{D}}_{\textrm{s}}$, $\bar {\textrm{D}}_{\textrm{i}}$ in ${\textrm{D}}_{\textrm{i}}$).

3. Sub-Poissonian Twin Beams: Experimental Characteristics

In the experiment, PATWBs with marginal mean photon numbers $\langle n_{\textrm{s}}\rangle$ and $\langle n_{\textrm{i}}\rangle$ ranging from 2.8 to 7.5 were reached after adding up to three photons into each beam (for the idler-beam characteristics, see the Supplement 1). According to the curves in Fig. 3(a) the post-selecting detectors with $\eta \approx 0.2$ add around 1.75 mean number of photons to the mean number 2.63 of photon pairs of the original twin beam TWB${}_{\textrm{p}}$ if no post-selecting photocounts are detected [solid curves in Fig. 3(a)]. On the other hand, the majority of photons come from the original twin beam TWB${}_{\textrm{p}}$ when spatial photon-pair correlations in twin beams TWB${}_{\textrm{s}}$ and TWB${}_{\textrm{i}}$ are exploited [dashed curves in Fig. 3(a)]. This means that the use of spatial photon-pair correlations in the post-selection practically eliminates additional noise photons in the fields to be added, which effectively results in the post-selection with a nearly ideal photon-number-resolving detector. Contrary to this, when spatial photon-pair correlations are ignored, the mean number of additional noise photons in the fields to be added can be estimated by the number of photons that effectively lose their twins in the post-selecting detection, i.e., by $(1-\bar {\eta }_{\textrm{s}}) \langle n^{\textrm{p}}_{\textrm{s}}\rangle \approx 1.75$ considering the signal field. The properties of the post-selecting detection also lead to different numbers of photons added into a PATWB per one post-selecting photocount: the detection that uses the spatial photon-pair correlations and behaves close to the ideal detection adds close to one mean photon number per detected photocount. Contrary to this, only approximately 0.8 mean photon number is added per one detected photocount for $\eta \approx 0.2$ [note the height of stairs in the solid and dashed curves in Fig. 3(a)].

 

Fig. 3. Signal-beam (a) mean photon number $\langle n_{\textrm{s}}\rangle$, (b) Fano factor $F_{n,{\textrm{s}}}$, and (c) mutual entropy $G_{n,{\textrm{s}}}$ and (d) noise-reduction parameter $R_{n}$ as they depend on post-selecting signal photocount number $\bar {c}_{\textrm{s}}$ assuming the post-selecting idler photocount number $\bar {c}_{\textrm{i}}$ fixed: $\bar {c}_{\textrm{i}} = 0$ (red $\circ$, $\Box$); 1 (green $\ast$, $\times$); 2 (blue $\triangle$, $\nabla$); and 3 (black $\diamond$). Experimental data are plotted as isolated symbols with error bars (usually smaller than the plotted symbols), solid and dashed curves originate in the model. Symbols $\circ$, $\ast$, $\triangle$, $\diamond$ ($\Box$, $\times$, $\nabla$), and solid (dashed) curves ignore (take into account) spatial photon-pair correlations. The quantum-classical border $F_{n} = 1$ in (b) is plotted as a dotted black horizontal line. The experimental errors were derived from the number of measurement repetitions determined for each PATWB using the probabilities $p^{\textrm{p}}$ in Table 1 and the overall number $1.2 \times 10^6$ of performed measurements.

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Systematic increase of sub-Poissonianity of the marginal beams is the most important feature of PATWBs quantified in Fig. 3(b) by the Fano factor $F_{n,{\textrm{s}}}$ of the signal beam. According to the curves in Fig. 3(b) detection of just one post-selecting photocount $\bar {c}_{\textrm{s}}$ leads to weakly sub-Poissonian photon-number distributions certified by $F_{n,{\textrm{s}}}<1$. The signal and idler beams with the Fano factor $F_{n} \approx 0.8$ were observed after detecting $\bar {c}_{\rm s,i} = 3$ post-selecting photocounts. Inclusion of spatial photon-pair correlations in the post-selecting detection even allowed us to observe the beams with $F_{n} \approx 0.6$ after registering $\bar {c}_{\rm s,i} = 2$ photocounts. We note that, according to Table 1, the sum of probabilities $p^{\textrm{p}}$ of post-selecting these PATWBs is around 2.5% and it drops down below 1% when the spatial photon-pair correlations are used.

Adding photons into a multi-mode TWB does not ideally influence quantum correlations between the photon-number fluctuations of the signal and idler beams. The measurement that takes into account the spatial photon-pair correlations confirms this: the experimental values of noise-reduction parameter $R_n \approx 0.2$ drawn in Fig. 3(d) accord with the value $R_n^{\textrm{TWB}} = 0.2 \pm 0.01$ belonging to the original TWB${}_{\textrm{p}}$. On the other hand, the values of $R_n \approx 0.5$ obtained for PATWBs generated without considering the spatial photon-pair correlations reflect considerable weakening of the photon-number correlations. We note that because the added photons increase the nonclassicality of the marginal beams, they also increase the nonclassicality of the whole PATWB, as discussed in detail in the Supplement 1.

Addition of photons into a TWB also causes gradual departure of the field from its multi-mode Gaussian form. The induced non-Gaussianity can be quantified by the mutual entropy $G_{n,{\textrm{s}}}$ of the signal beam with its photon-number distribution $p_{\textrm{s}}(n_{\textrm{s}})$ [61],

4. Sub-Poissonian Twin Beams: Microscopic View

Finally, we provide a “microscopic” view on the process of photon addition into a TWB as revealed by the quantum theory of light that describes a PATWB by its joint quasi-distribution $P_{\textrm{si}}$ of integrated intensities. In the quantum theory and considering multi-mode optical fields (the modes phases play minimal role), an optical field is described in general by a proper quasi-distribution of integrated intensities [1]. Assigning to the analyzed multi-mode PATWBs their joint signal-idler quasi-distributions $P_{\textrm{si}}(W_{\textrm{s}},W_{\textrm{i}})$ of integrated intensities $W_{\textrm{s}}$ and $W_{\textrm{i}}$ given for the normal field-operator ordering [15], their joint signal-idler photon-number distributions $p_{\textrm{si}}(n_{\textrm{s}},n_{\textrm{i}})$ are obtained by applying the Mandel detection formula:

Inversion of the generalized Mandel detection formula then allows us to determine the quasi-distributions $P_{\textrm{si}}(W_{\textrm{s}},W_{\textrm{i}};s)$ for the general $s$-ordered field operators along the formula [15]:

The quasi-distributions $P_{\textrm{si}}(W_{\textrm{s}},W_{\textrm{i}};s)$ of PATWBs with zero and one pair of photocounts added by using the spatial photon-pair correlations are compared in Fig. 4, together with their photon-number distributions $p_{\textrm{si}}(n_{\textrm{s}},n_{\textrm{i}})$. The quasi-distribution $P_{\textrm{si}}$ of PATWB with no added photocounts plotted in Fig. 4(b) is very close to that of the original TWB with the characteristic negative fan-shaped areas. Addition of photon pairs then emphasizes the central positive part of the quasi-distribution $P_{\textrm{si}}(W_{\textrm{s}},W_{\textrm{i}};s)$ which is accompanied by the deformation of the original fan-shaped negative areas, as demonstrated in Fig. 4(d). We note that the negative values of quasi-distributions $P_{\textrm{si}}$ drawn in Figs. 4(b) and 4(d) for the ordering parameter $s = -0.1$ mean that the nonclassicality depths $\tau$ of these PATWBs have to be greater than 0.55 ($\tau = (1-s)/2$, see Supplement 1 and Ref. [63]). We also note that the number of negative areas as well as the values of negative quasi-distributions $P_{\textrm{si}}$ strongly depend on the field-operator ordering parameter $s$: the closer to 1 the value of the ordering parameter $s$ is, the greater the number of negative areas is and also the more negative the values of quasi-probabilities $P_{\textrm{si}}$ are [15].

 

Fig. 4. (a),(c) Photon-number distribution $p_{\textrm{si}}$ and (b),(d) quasi-distribution $P_{\textrm{si}}$ of integrated intensities (field-operator ordering parameter $s=-0.1$) drawn for PATWBs with (a),(b) $\bar {c}_{\textrm{s}} = \bar {c}_{\textrm{i}} = 0$ and (c),(d) $ \bar{c}_{\rm s} = \bar{c}_{\rm i} = 1 $ considering the spatial photon-pair correlations. In white areas in (b),(d), probability density $P_{\textrm{si}}$ is negative.

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5. Conclusions

We have demonstrated that addition of photons into multi-mode twin beams results in their marginal sub-Poissonian photon-number distributions while it ideally keeps their quantum photon-number cross-correlations. Using three twin beams each with around 2.5 mean photon-pair number and post-selection based on photon-number-resolving detection by an iCCD camera, we have experimentally generated sub-Poissonian twin beams with Fano factors $F_n\approx 0.8$ and verified their quantum photon-number cross-correlations. Taking into account tight spatial photon-pair correlations, that allow us to mimic the behavior of ideal photon-number-resolving detection, the beams with Fano factors $F_n\approx 0.6$ were even reached. The increase of non-Gaussianity as well as nonclassicality of these beams with the increasing number of added photons has been experimentally demonstrated.

The multi-mode sub-Poissonian twin beams are prospective for quantum measurements of two-photon absorption cross-sections that are used in the virtual-state entangled-photon spectroscopy. Sub-Poissonianity of the beams, similarly as their spectral and spatial multi-modality, allow us to considerably reduce the number of measurements of two-photon absorption cross-sections, which moves the virtual-state entangled-photon spectroscopy closer to its practical implementation. Unusual statistical properties make these beams in general attractive for engineered two-photon excitations of atom, molecular, and other material systems under specific conditions.