1. Introduction

States with physical quantities whose fluctuations are suppressed below the shot-noise limit [1] given by the coherent states [2] are attractive for quantum metrology. Sub-Poissonian states [3] of an optical beam and states of twin beams (TWBs) [4–9] whose difference of photon-number fluctuations lies below the shot-noise level belong to such states. Whereas the sub-Poissonian beams allow quantum measurements of single-photon absorption, TWBs are attractive for monitoring two-photon absorption processes [10]. The measurement of two-photon absorption cross-sections by entangled optical beams [11] performed for varying entanglement times and relative beam delays has even allowed to develop a new form of two-photon spectroscopy called the virtual-state entangled-photon spectroscopy [12] that enables in principle to reveal the complete sample spectrum. This method was originally suggested for entangled photon-pair states that, however, suffer from low photon-pair fluxes that require huge dipole moments in real samples. Later, multi-mode TWBs with photon-pair fluxes greater by many orders in magnitude have been shown in [13–15] as alternative sources for this spectroscopy. Nevertheless, marginal photon-number distributions of both their beams are close to Poissonian, i.e., classical. This suggests to look for two-beam states that are endowed, on the top of spectral correlations, both with strong correlations in photon-number fluctuations of the beams and marginal sub-Poissonian photon-number distributions, at least in one beam. Such states may bring the advantage of very high precision of the quantum measurement of two-photon absorption cross-sections. This would further advance the methods for determination of two-photon absorption cross-sections including the virtual-state entangled-photon spectroscopy.

Looking at the history of sub-Poissonian light generation, we find an inspiration. The best performing method for sub-Poissonian-light generation is based upon post-selection by photon-number-resolving detection applied to a TWB [16–18]. It has been applied using both cw [16,19,20] and pulsed [7,17,21–24] TWBs. The properties of sub-Poissonian states generated this way have been addressed from different points of view including higher-order photon-number squeezing and nonclassicality [18,24–28]. Their application in quantum, i.e. sub-shot-noise, measurements of absorption [29,30] and quantum communications [31] has been suggested. The former one has been experimentally demonstrated and developed into the form of imaging systems [32–38] and spectrometers [39].

Inspired by the above cited post-selection method, we suggest to divide a TWB into two parts with their own photon-number correlations. The first part is kept as it is and guarantees strong (ideal) photon-number correlations of the resultant state. The remaining part is then used in the post-selection process to give rise to a field with sub-Poissonian photon-number distribution (for the scheme, see Fig. 1). Both parts, when considered together, form a state with both strong photon-number correlations between the beams and sub-Poissonian photon-number distribution in one beam - the generalized sub-Poissonian state. The word generalized refers to the fact that the field exhibits not only suppression of the beam photon-number fluctuations below the vacuum level but also strong (quantum) inter-beam photon-number correlations. The presence of quantum correlations in these states in fact leads to a new group of nonclassical states with specific properties.

 

Fig. 1. Scheme for generating sub-Poissonian states of two-beam fields: The signal beam of a twin beam TWB emitted in a nonlinear crystal is partially reflected at beam splitter BS${}_{\rm s}$ with transmissivity $t_{\rm s}$ and its reflected part is detected at detector $\bar {\rm D}_{\rm s}$ serving for post-selection. The transmitted signal beam and the whole idler beam are left, after suitable post-selection, in the states with sub-Poissonian marginal idler fields. These fields are monitored at detectors ${\rm D}_{\rm s}$ and ${\rm D}_{\rm i}$. Each detector is characterized by its detection efficiency $\eta$, dark-count rate per pixel $d$ and number $N$ of pixels. Photon numbers of the whole signal ($\tilde {n}_{\rm s}$), reflected ($\bar {n}_{\rm s}$) and transmitted ($n_{\rm s}$) signal and the whole idler ($n_{\rm i}$) beams are shown.

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Division of one beam from the TWB into two parts and its subsequent partial detection in fact represents the mechanism of photon subtraction, suggested in [40] for a single-mode optical field. The comprehensive analysis of single-mode photon-subtracted fields performed in [41] revealed, among others, that direct photon subtraction from a state with classical photon-number statistics does not transform this state into a sub-Poissonian one. Different situation occurs when photons are subtracted from a two-mode squeezed vacuum state [42], which represents a specific case of a TWB with just one mode in the signal beam and one mode in the idler beam. In such case, the sub-Poissonian photon-number distribution may be reached in principle in the complementary beam, in which the post-selection measurement is not performed. However, photon subtraction from a two-mode squeezed vacuum state has been found experimentally too difficult in [42] to reach the sub-Poissonian beam. The reason is that the marginal photon-number distributions of a two-mode squeezed vacuum state are single-mode thermal, i.e. they are far from the Poissonian classical-quantum border. Though the marginal photon-number fluctuations of the states were decreasing with the increasing number of subtracted photons, the Poissonian classical-quantum border was not reached in [42].

The use of a multi-mode TWB as the state for photon subtraction represents the key for reaching the marginal sub-Poissonian photon-number distributions and thus generating the generalized sub-Poissonian states. The structures of a two-mode squeezed vacuum state and a multi-mode TWB are modified qualitatively differently by photon subtraction. The reason is that the multi-mode TWB is composed of a larger number of independent spatio-spectral modes and a photon is removed from an arbitrary mode with the same probability. The mechanism of photon subtraction thus introduces certain photon-number correlations among the modes. We may have a look at photon subtraction from the multi-mode TWB formally: Its multi-mode structure results in the marginal photon-number distribution being nearly Poissonian. Then even a small reduction of photon-number fluctuations by photon subtraction naturally results in the sub-Poissonian distributions. Here, we report on a successful experimental generation of these states using a multi-mode TWB [43] and post-selection based on the detection of a given number of photocounts by an intensified CCD (iCCD) camera [44].

The paper is organized as follows. Theoretical properties of generalized sub-Poissonian states are analyzed in Sec. 2. In Sec. 3, experimental generation of these states is discussed using their basic characteristics. Their nonclassicality is studied in Sec. 4. Entropy, coherence properties and non-Gaussianity of these states are analyzed in Sec. 5. The use of spatial photon-pair correlations in the generated fields to improve their nonclassical properties is discussed in Sec. 6. Sec. 7 brings conclusions.

2. Theoretical considerations

To reveal the role of multi-modality in obtaining sub-Poissonian photon-number distributions by photon subtraction, we first consider a simple model of an $M_{\rm p}$-mode noiseless TWB having on average $B_{\rm p}$ photon pairs in each mode. We note that we consider here the equally populated modes as it is known from the classical coherence theory [45] that, in the case of the fields composed of greater numbers of independent modes with comparable properties, the parameters of individual modes influence only negligibly the properties of the overall fields. The photon-number distribution $p^{\rm TWB}_{\rm id}$ of a noiseless TWB is expressed by the Mandel-Rice formula [46] that gives $p^{\rm TWB}_{\rm p}$ written below (for details about constructing models of photon-number distributions, see [3,47,48]):

Quantifying the idler-beam sub-Poissonianity by the Fano factor $F_{n,{\rm i}}<1$ [3,46] and quantum correlations between the signal- and idler-beam photon-numbers by the noise-reduction parameter $R_n < 1$ [4–9],

 

Fig. 2. (a,b,d,e) Idler-beam Fano factor $F_{n,{\rm i}}$ and (c,f) noise-reduction parameter $R_{n}$ as they depend on beam-splitter transmissivity $t_{\rm s}$ and post-selecting photocount number $\bar {c}_{\rm s}$ considering (a,d) single-mode TWB with $M_{\rm p} = 1$ and $B_{\rm p} = 10$ and (b,c,e,f) multi-mode TWB with $M_{\rm p} = 100$ and $B_{\rm p} = 0.1$; $\bar {\eta }_{\rm s} = 1$ (a-c) and $\bar {\eta }_{\rm s} = 0.25$ (d-f). In (a,b,d,e) the quantum-classical border $F_{n,{\rm i}} = 1$ is indicated by dashed black curves, $F_{n,{\rm s}} > 1$ in the white area.

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Correlations between the signal- and idler-beam photon numbers are more robust than sub-Poissonianity and strongly quantum-correlated two-beam states are obtained both for single- and multi-mode TWBs as well as for the ideal [Fig. 2(c)] and realistic [Fig. 2(f)] detection. We note that with $\bar {\eta }_{\rm s} = 1$ [Fig. 2(c)], the noise-reduction parameter $R_n$ equals 0 for any value of $t_{\rm s}$, as the subtraction of a given number $\bar {c}_{\rm s}$ of photocounts in the signal beam in fact means the subtraction of the same number of photons from the signal beam and we then have $n_{\rm i} - n_{\rm s} = \bar {c}_{\rm s}$ together with $\langle n_{\rm i}\rangle - \langle n_{\rm s}\rangle = \bar {c}_{\rm s}$. This means that the correlations of photon-number fluctuations between the signal and idler beams remain ideal.

Excluding a small area around $t_{\rm s} = 1$ [see Figs. 2(a,d)] it holds in general that the increasing values of the beam-splitter transmissivity $t_{\rm s}$ lead to stronger photon-number correlations (smaller $R_n$) but they worsen the conditions for sub-Poissonian-light generation (greater $F_{n,{\rm i}}$).

3. Experimental generalized sub-Poissonian fields: basic properties

The properties of investigated states as revealed by the above simple model were studied in the experimental setup shown in Fig. 3(a). A 5-mm-long type-I beta-barium-borate crystal (BaB${}_2$O${}_4$, BBO) cut for a slightly non-collinear geometry provided the multi-mode TWB used for photon subtraction. Spontaneous parametric down-conversion was pumped by the third-harmonic pulses (280 nm) generated by the pulses of a femtosecond cavity dumped Ti:sapphire laser (840 nm, 150 fs). The signal and idler beams (at nearly-frequency-degenerated frequencies $\approx$560 nm) were filtered by a 14-nm-wide bandpass interference filter. We note that the width of interference filter in fact defines the spatial profile of the used TWB and it is chosen such that suitable balance of intensity and photon-number correlations in the TWB is reached. The signal beam was split in its transverse plane into two parts to mimic the behavior of beam splitter BS${}_{\rm s}$ with transmissivity $t_{\rm s}$ considered in the scheme in Fig. 1 for generating the generalized sub-Poissonian states. The analyzed three beams were detected in different detection areas at the photocathode of an iCCD camera Andor DH334-18U-63, as shown in Fig. 3(b). Whereas the detection area $\bar {\rm D}_{\rm s}$ [$\bar {N_{\rm s}}$, $\bar {d_{\rm s}} = d_{\rm s}$] monitored the photocounts used in post-selecting the generalized sub-Poissonian states, the remaining two detection areas ${\rm D}_{\rm s}$ [$N_{\rm s} + \bar {N_{\rm s}} = 4536$, $(N_{\rm s} + \bar {N_{\rm s}}) d_{\rm s} = 0.22$] and ${\rm D}_{\rm i}$ [$N_{\rm i} = 4536$, $d_{\rm i}N_{\rm i} = 0.22$] measured the photocounts belonging to the analyzed states. The performed experiment provided us with the 3D photocount histogram $f(c_{\rm s},c_{\rm i};\bar {c}_{\rm s})$ that involves 2D photocount histograms of generalized sub-Poissonian states obtained upon registering a fixed number $\bar {c}_{\rm s}$ of the post-selecting photocounts.

 

Fig. 3. (a) Experimental setup: A nonlinear crystal BBO is used to generate TWBs by ultra-short third-harmonic pulses with actively stabilized intensity using rotating half-wave plate HWP, polarizing beam splitter PBS, and detector D as an active feedback. The signal beam is spatially divided in its transverse plane into two parts which mimics beam splitter BS${}_{\rm s}$ in Fig. 1. The beams, after passing through bandpass interference filter F, are detected by an iCCD camera. (b) Photocathode of an iCCD camera showing the multiply exposed signal and idler detection strips. Detection areas covering the signal and idler detection strips and playing the role of detectors ${\rm D}_{\rm s}$, $\bar {\rm D}_{\rm s}$, and ${\rm D}_{\rm i}$ are defined. 3D photocount histogram $f(c_{\rm s},c_{\rm i};\bar {c}_{\rm s})$ is constructed counting the photocounts in three indicated detection areas in $1.2\times 10^6$ repetitions of the measurement.

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We note that the partitioning of the signal beam at the photocathode of iCCD camera does not directly correspond to the geometry in which a beam splitter with transmissivity $t_{\rm s}$ is put into the signal beam beyond the nonlinear crystal (see Fig. 1). This is caused by spatial correlations between the signal and the idler beams in their transverse planes that are described by the correlated areas [44] smaller than the detection areas. Nevertheless, we may imagine that the random process of choosing an output port of the beam splitter by a signal photon is realized already in the process of parametric down-conversion in the nonlinear crystal. Though the accompanying idler photon bears the information about the spatio-spectral mode chosen by the signal twin, this information is concealed by ignoring the positions of photocounts in the idler beam when processing the experimental data. We note that, by using a suitable optical imaging system, the signal and the idler beams can be transformed into specific transverse planes [50,51] in which they lose their spatial correlations. When detecting the beams in these planes, the one-to-one correspondence between this experiment and the experiment with the beam splitter beyond the nonlinear crystal is reached.

The experimental 2D photocount histograms $f(c_{\rm s},c_{\rm i};\bar {c}_{\rm s})$ characterizing the generalized sub-Poissonian states post-selected by detecting $\bar {c}_{\rm s}$ photocounts were used in the maximum-likelihood reconstruction method [43,52] to arrive at the corresponding photon-number distributions $p(n_{\rm s},n_{\rm i};\bar {c}_{\rm s})$. The detection efficiencies $\eta$ belonging to the used detection areas ${\rm D}_{\rm s}$, ${\rm D}_{\rm i}$, and $\bar {\rm D}_{\rm s}$ were determined using the detector calibration technique developed in [53]: $\eta _{\rm s} = \bar {\eta _{\rm s}} = 0.234\pm 0.005$, $\eta _{\rm i}=0.227\pm 0.005$. The calibration method considers the photon-number distribution $p^{\rm TWB}$ of the used noisy TWB in the form of two-fold convolution of the ideal photon-pair-number distribution $p^{\rm TWB}_{\rm id}$ written in Eq. (1) with the signal ($\tilde {p}_{\rm n,s}$) and the idler ($p_{\rm n,\ i}$) noise photon-number distributions. The latter describe the fields composed of $\tilde {M}_{\rm s}$ signal and $M_{\rm i}$ idler modes each populated by mean $\tilde {B}_{\rm s}$ signal and $B_{\rm i}$ idler photons. The convolution is written as follows:

Both noise distributions in Eq. (5) are assumed in the Mandel-Rice form $p_{{\rm n}}(n;M,B) = \Gamma (n+M) / [n! \Gamma (M)] B^{n} /(1+B)^{n+M}$ of the field with $M$ modes each populated by mean $B$ photons. The detector calibration method gave the best fit to the experimental TWB assuming $M_{\rm p}=168$, $B_{\rm p}=0.048$, $\tilde {M}_{\rm s}=0.008$, $\tilde {B}_{\rm s}=8.8$, $M_{\rm i} = 0.020$, and $B_{\rm i}=6.3$ (relative experimental errors: 7%, for more details, see [47]). Thus, the TWB was composed of on average 8.0 photon pairs and 0.07 (0.16) noise signal (idler) photons. We note that the experiment was repeated $1.2\times 10^6$ times. The experimental errors drawn in the graphs are the statistical errors derived just from this number of measurement repetitions. As such they give the lowest estimate for real errors.

In the experiment, three values of transmissivity $t_{\rm s}$ were used: Whereas the lowest value $t_{\rm s} = 0.190$ favors sub-Poissonianity of the generated states, the greatest value $t_{\rm s} = 0.805$ prefers sub-shot-noise photon-number correlations between the beams. The middle value $t_{\rm s} = 0.505$ then proportionally combines both nonclassical properties. Up to four photocounts ($\bar {c}_{\rm s}$) were subtracted to arrive at the generalized sub-Poissonian states (see Fig. 4). The way of generation naturally leads to greater idler photon numbers compared to the signal ones. Moreover, the signal mean photon numbers $\langle n_{\rm s}\rangle$ practically do not depend on the post-selecting photocount number $\bar {c}_{\rm s}$, contrary to the idler mean photon numbers $\langle n_{\rm i}\rangle$ that increase roughly linearly with $\bar {c}_{\rm s}$, as shown in Figs. 4(a,b). The applied method also allows for the generation of sub-Poissonian light only in the idler beam provided that the transmissivity $t_{\rm s}$ is sufficiently small and the post-selecting photocount number $\bar {c}_{\rm s}$ is sufficiently large [see Figs. 4(c,d)].

 

Fig. 4. (a) [(b)] Mean signal- [idler-] beam photon number $\langle n_{\rm s}\rangle$ [$\langle n_{\rm i}\rangle$] and (c) [(d)] Fano factor $F_{n,{\rm s}}$ [$F_{n,{\rm i}}$ ], (e) noise-reduction parameter $R_n$ and (f) probability $p^p$ of state generation as they depend on post-selecting photocount number $\bar {c}_{\rm s}$ for $t_{\rm s} = 0.190$ (red $\circ$), 0.505 (green $\ast$) and 0.805 (blue $\triangle$). Data considering spatial photon-pair correlations and $t_{\rm s} = 0.190$ are plotted using black $\diamond$ (for details, see Sec. 6). Experimental data are plotted as isolated symbols with error bars, solid curves originate in the model. In (c,d), the quantum-classical border $F_{n} = 1$ is plotted as a dashed black horizontal line. The number of measurement repetitions for given $\bar {c}_{\rm s}$, that determines the experimental errors, is derived from the corresponding probability $p^p$ drawn in (f) considering $1.2 \times 10^{6}$ overall measurement repetitions.

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The noise-reduction parameter $R_n$ defined in Eq. (5) changes only weakly with the increasing post-selecting photocount number $\bar {c}_{\rm s}$ provided that the transmissivity $t_{\rm s}$ is large, i.e. when the mean photon numbers $\langle n_{\rm s}\rangle$ and $\langle n_{\rm i}\rangle$ are comparable [see the blue curve and $\triangle$ in Fig. 5(e)]. The model of an ideal noiseless TWB and post-selection by a detector without dark counts suggests a small decrease of $R_n$ with the increasing $\bar {c}_{\rm s}$ [see Fig. 2(f)]. As the mean number of signal photons illuminating the post-selecting detector $\bar {\rm D}_{\rm s}$ is small in this case, dark counts of the post-selecting detector play a significant role and they are responsible for the behavior of $R_n$ observed in Fig. 4(e). On the other hand, if the transmissivity $t_{\rm s}$ is small, the mean photon numbers $\langle n_{\rm s}\rangle$ and $\langle n_{\rm i}\rangle$ considerably differ and the noise-reduction parameter $R_n$ decreases [see the red curve and $\circ$ in Fig. 4(e)]. This accords with the model of an ideal noiseless TWB and post-selection by a detector without dark counts [see Fig. 2(f)]. Larger mean numbers of signal photons illuminating the post-selecting detector suppress the role of detector dark counts in this case.

 

Fig. 5. (a) [(b)] Nonclassicality depth $\tau _M$ [$\tau _{L,{\rm i}}$] of generalized sub-Poissonian states [their idler beam] as it depends on post-selecting photocount number $\bar {c}_{\rm s}$ for $t_{\rm s} = 0.190$ (red $\circ$), 0.505 (green $\ast$) and 0.805 (blue $\triangle$). Data considering spatial photon-pair correlations and $t_{\rm s} = 0.190$ are plotted using black $\diamond$ (see Sec. 6). Experimental data are plotted as isolated symbols with error bars (for details, see the caption to Fig. 4), solid curves originate in the model.

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We note that the values of the experimental noise-reduction parameter $R_n$ drawn in Fig. 4(e) for $t_{\rm s} = 0.190$ are systematically greater than their theoretical counterparts. This is caused by weaker photocount correlations at the edges of the strips compared to their central parts; they give the main contribution in this case of small transmissivity $t_{\rm s}$ [see Fig. 3(b)]. This systematic declination is also observed in the lower experimental values of the nonclassicality depth $\tau _M$ plotted in Fig. 5(a).

On the other hand, the model gives considerably greater values of the signal-beam Fano factor $F_{n,{\rm s}}$ and the noise-reduction parameter $R_n$ for $t_{\rm s} = 0.805$ [see Figs. 4(c,e)]. This is caused by overestimating the level of dark counts $\bar {d}_{\rm s}$ of the post-selecting detector $\bar {\rm D}_{\rm s}$ and high sensitivity of $F_{n,{\rm s}}$ and $R_n$ to the dark-count level under the studied conditions (low detector illumination, greater post-selecting photocount numbers $\bar {c}_{\rm s}$).

Probabilities $p^p$ of generating the generalized sub-Poissonian states are plotted in Fig. 4(f). They show that the states with up to three subtracted photocounts dominate among the generated states. We note that Fig. 4 also contains the data obtained by using spatial photon-pair correlations that are introduced and discussed in Sec. 6 below.

4. Nonclassicality of generalized sub-Poissonian fields

Both the idler-beam sub-Poissonianity and sub-shot-noise photon-number correlations contribute to the nonclassicality of the emitted generalized sub-Poissonian fields. This nonclassicality can be quantified using the Lee nonclassicality depth [54] provided that a suitable nonclassicality inequality serving as a nonclassicality identifier (NCI) for a given type of states exists. We have tested the performance of the NCIs used for revealing the nonclassicality of TWBs in [55] and found the following suitable NCI $M$:

In Eq. (6), a $k$th-order intensity moment $\langle W^k\rangle$ is the $k$th-order normally-ordered photon-number moment determined as $\langle W^k\rangle = \sum _{l=1}^{k} S(k,l) \langle n^l\rangle$ [48] using the Stirling numbers $S(k,l)$ of the first kind [56]. Similarly, the NCI $L$ is suitable for revealing the local nonclassicality originating in sub-Poissonianity in the idler beam:

According to the graphs in Fig. 5(a), the nonclassicality depth $\tau _M$ of the generalized sub-Poissonian states slowly decreases with the increasing post-selecting photocount number $\bar {c_{\rm s}}$ provided that the transmissivity $t_{\rm s}$ is large (blue $\triangle$). In this case, photon pairs prevail in the generated states and the additional idler photons originating in the post-selection unbalance the signal- and idler-beam photon numbers which results in the gradual loss of the nonclassicality. On the other hand, if the transmissivity $t_{\rm s}$ is smaller the nonclassicality depth $\tau _M$ slowly increases with the increasing $\bar {c_{\rm s}}$. The reason is that the number of photon pairs in the state is smaller and so the additional idler photons from the post-selection process considerably modify the states and lead to sub-Poissonian idler-beam photon-number distributions that contribute both to the local [see Fig. 5(b)] and global nonclassicalities [Fig. 5(a)]. From the point of view of the varying transmissivity $t_{\rm s}$, the global ($\tau _M$) and local ($\tau _{L,{\rm i}}$) nonclassicality depths behave in the complementary way: Whereas the global nonclassicality depth $\tau _M$ increases with the increasing transmissivity $t_{\rm s}$, the local nonclassicality depth $\tau _{L,{\rm i}}$ decreases [compare Figs. 5(a,b)]. It follows from the comparison of Figs. 4(d) and 5(b) that the greater the nonclassicality depth $\tau _{L,{\rm i}}$ is, the smaller the Fano factor $F_{n,{\rm i}}$ is.

5. Entropy, mutual photon-number coherence and non-Gaussianity

Subtraction of photons can be seen as addition of photons into the other (idler) beam (see the scheme for photon addition in [41]). So, it adds photons into the idler beam that gradually increase the difference between the idler- and signal-beam photon numbers. This leads to the gradual but slow increase of entropy $S$ of the generalized sub-Poissonian states,

 

Fig. 6. (a) Entropy $S$ and (b) degree $\gamma _{\rm si}$ of mutual photon-number coherence as they depend on post-selecting photocount number $\bar {c}_{\rm s}$ for $t_{\rm s} = 0.190$ (red $\circ$), 0.505 (green $\ast$) and 0.805 (blue $\triangle$). Data considering spatial photon-pair correlations and $t_{\rm s} = 0.190$ are plotted using black $\diamond$ (see Sec. 6). Experimental data are plotted as isolated symbols with error bars (for details, see the caption to Fig. 4), solid curves originate in the model.

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Adding photons into the idler beam drives the field out of its original Gaussian form, which increases its nonclassicality. The greater the number of subtracted signal photocounts $\bar {c}_{\rm s}$ is the greater the idler-beam declination from its Gaussian form is [see Fig. 7(a)]. Declination from non-Gaussianity is monitored by the mutual entropy $G_{n,{\rm i}}$ [57] that uses the beam photon-number distribution $p_{\rm i}(n)$ and its Gaussian reference $p^{\rm MR}(n)$ in the multi-mode thermal Mandel-Rice form [48] with the same mean photon numbers:

Increasing non-Gaussianity of the idler beam induces increasing non-Gaussianity of the whole photon-subtracted field. This non-Gaussianity may also be quantified by the mutual entropy $G_n$ defined in analogy to Eq. (10). In this case, we use as a reference a specific multi-mode two-beam Gaussian field containing both photon pairs and noisy photons [53,58]. The values of the mutual entropy $G_n$ minimized over the reference-field parameters and plotted in Fig. 7(b) confirm this behavior.

 

Fig. 7. Mutual entropies (a) $G_{n,{\rm i}}$ of the idler beam and (b) $G_{n}$ of the whole field as they depend on post-selecting signal photocount number $\bar {c}_{\rm s}$ for $t_{\rm s} = 0.190$ (red $\circ$), 0.505 (green $\ast$) and 0.805 (blue $\triangle$). Data considering spatial photon-pair correlations and $t_{\rm s} = 0.190$ are plotted using black $\diamond$ (see Sec. 6). Experimental data are plotted as isolated symbols with error bars (for details, see the caption to Fig. 4), solid curves originate in the model.

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6. Improvement by spatial photon-pair correlations

The use of an iCCD camera as a photon-number-resolving detector allows to exploit tight spatial correlations [30,44] between the photons in a photon pair to subtract only the photocounts of photon pairs with both photons detected. This considerably decreases the success probability of generating the generalized sub-Poissonian states [see black $\diamond$ and curve in Fig. 4(f)], but it substantially improves the quantum properties of the whole fields as well as their idler beams. We analyze only the case with $t_{\rm s} = 0.190$ that gives comparable mean (spatially correlated) photocount numbers at both signal-beam detectors. We note that the consideration of spatial photon-pair correlations effectively decreases the mean photocount number $\langle \bar {c}_{\rm s} \rangle$ at the post-selecting detector $\bar {\rm D}_{\rm s}$ $1/\eta _{\rm i}$ times. Thus, for $\eta _{\rm i} \approx 0.2$ the effective mean photocount number $\langle \bar {c}_{\rm s}\rangle$ at the post-selecting detector drops down about 5 times. On the other hand, for $t_{\rm s} = 0.190$, the mean photocount number $\langle \bar {c}_{\rm s}\rangle$ at the post-selecting detector is about 4 times greater than the mean photocount number $\langle c_{\rm s}\rangle$ of the analyzed signal beam.

The post-selecting mechanism keeps the mean signal photon number $\langle n_{\rm s}\rangle$ constant while increasing the mean idler photon number $\langle n_{\rm i}\rangle$ by one per each subtracted photocount numbered by $\bar {c}_{\rm s}$, as shown in Figs. 4(a,b) using the experimental data (black $\diamond$ and curves). Compared to the usual subtraction scheme the inclusion of spatial correlations considerably reduces the idler photon numbers $\langle n_{\rm i}\rangle$ by eliminating the unpaired idler-beam photocounts in the corresponding idler-beam detection area. Inclusion of spatial photon-pair correlations results in large reduction of both the idler-beam Fano factor $F_{n,{\rm i}}$ and noise-reduction parameter $R_n$, as evidenced in Figs. 4(d,e). According to Fig. 4(d), the lowest experimental value $F_{n,{\rm i}} = 0.47 \pm 0.14$ of the idler-beam Fano factor is reached for $\bar {c}_{\rm s} = 2$ using this subtraction scheme. Inclusion of spatial photon-pair correlations naturally leads to considerably greater nonclassicality depths $\tau _{L,{\rm i}}$ and $\tau _M$ of the idler beam and the whole field, respectively, as shown in Fig. 5 (compare red $\circ$ with black $\diamond$). Elimination of the unpaired idler-beam photocounts discussed above reduces the entropy $S$ that does not change with $\bar {c}_{\rm s}$, as experimentally confirmed in Fig. 6(a). It also naturally increases the degree $\gamma _{\rm si}$ of mutual photon-number coherence that, however, decreases with the increasing $\bar {c}_{\rm s}$. This is a consequence of increasing unbalance between the signal and idler mean photon numbers [see Fig. 6(b)]. The missing unpaired idler-beam photocounts cause considerable departure of the analyzed states from the Gaussian ones, both when the whole fields and their idler beams are analyzed, as documented in Fig. 7.

7. Joint photon-number distributions, integrated-intensity quasi-distributions and local nonclassicality

Eventually, we address in detail the properties of generalized sub-Poissonian states. In Fig. 8, the photon-number distributions $p$ and the corresponding quasi-distributions $P$ of integrated intensities are plotted (for their determination, see [59]) for two typical states: A state with the prevailing correlations between the signal- and idler-beam photon numbers [Figs. 8(a,b)] and a state exhibiting strong sub-Poissonianity in the idler beam [Figs. 8(d,e)]. The signal- and idler-beam photon numbers are comparable in the former state which results in the roughly symmetric intensity quasi-distribution $P$ resembling that of a TWB [60]. On the other hand, the idler-beam photon numbers $n_{\rm i}$ are about three-times larger than the signal-beam photon numbers $n_{\rm s}$ in the latter state which leads to the strongly asymmetric intensity quasi-distribution $P$. In both cases, we plot in Figs. 8(b,e) the intensity quasi-distributions $P$ close to the border between the quantum and classical forms of the quasi-distributions (field-operator ordering parameter $s= 0.07$). This gives us an estimate for the state nonclassicality depths $\tau =0.465$.

 

Fig. 8. (a,d) Photon-number distribution $p(n_{\rm s}, n_{\rm i})$, (b,e) quasi-distribution $P_s(W_{\rm s},W_{\rm i})$ of integrated intensities, and (c,f) local nonclassicality depth $\bar {\tau }_{\bar {C}_p}(n_{\rm s},n_{\rm i})$ plotted for generalized sub-Poissonian states with (a-c) $t_{\rm s} = 0.805$ and $\bar {c}_{\rm s} = 0$ and (d-f) $t_{\rm s} = 0.190$ and $\bar {c}_{\rm s} = 4$. In (b,e), s = 0.07; in (c,f), $\bar {C}_p$ is determined provided that the mean of the used probabilities is greater than 0.02.

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The distribution of nonclassicality across the photon-number distributions $p$ of these states is monitored in Figs. 8(c,f) via the nonclassicality depth $\bar {\tau }_{\bar {C}_p}$ belonging to the NCIs $\bar {C}_p$ defined as [61]:

According to Figs. 8(c,f), the maximal values of nonclassicality depths $\bar {\tau }$ are close to 0.4. Comparing the obtained values of nonclassicality depths, we may order the NCIs according to the increasing ability to quantify the nonclassicality in the analyzed states as follows: intensity-moment NCIs (see Fig. 5), probability NCIs [Fig. 8(c,f)], and integrated-intensity quasi-distributions with the varying ordering parameter [Fig. 8(b,e)].

8. Conclusion

By subtraction of fixed numbers of photocounts from one beam of a multi-mode twin beam we have obtained the generalized sub-Poissonian states. These states combine sub-Poissonianity of one beam with sub-shot-noise photon-number correlations between both beams. Dependence of these properties on the conditions of generation was experimentally investigated using the corresponding Fano factors and noise-reduction parameters. High level of their nonclassicality was certified by applying suitable nonclassicality identifiers as well as by determining the quasi-distributions of their integrated intensities. The use of spatial photon-pair correlations to enhance nonclassical features of the generalized sub-Poissonian states was demonstrated. The generalized sub-Poissonian states are prospective for quantum measurements of two-photon absorption cross-sections, sub-shot-noise determination of phases, and realization of engineered two-photon excitations of atomic, molecular, and other material systems. They give a promise for generalization of virtual-state entangled-photon spectroscopy exploiting their lower photon-number fluctuations compared to those of the usual twin beams.