Open Access
Add to BibSonomy
Abstract
Photon-number-resolved post-selection on one beam out of a correlated system of three beams with bi-partite photon-number correlations gives rise to joint photon-number distributions with the probabilities forming checkered patterns. These patterns originate in the convolution of two constituting photon-number distributions, one endowed with correlations in photon numbers, the other exhibiting anti-correlations in photon-number fluctuations. Using three twin beams of comparable intensity whose constituting beams suitably overlap on the photocathode of a photon-number-resolving iCCD camera, we experimentally as well as theoretically analyze the properties of such states as they change with the varying ratio of the correlated and anti-correlated contributions. The experimental photocount 2D histograms of the fields post-selected by the iCCD camera that are reconstructed by the maximum-likelihood approach confirm their non-classicality though the limited detection efficiency in post-selection conceals the checkered patterns. Contrary to this, the maximum-likelihood reconstruction of the experimental 3D photocount histogram similarly as a suitable 3D Gaussian fit, that reveal the states as they would be obtained by ideal post-selection, provide the photon-number distributions with the checkered patterns. The corresponding quasi-distributions of integrated intensities are determined. Nonclassical properties of the generated states are investigated using suitable non-classicality criteria and the corresponding non-classicality depths. These states with their correlations of varying intensity are prospective for two-photon excitations of atoms and molecules as well as two-photon spectroscopy.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Progress in the construction of photon-number-resolving detectors (PNRDs) [based on temporal multiplexing [1–3], spatial multiplexing [4–10], super-conducting bolometers [11–13], silicon matrix photodetectors [14,15], etc.] has stimulated fast development in the area of experimental investigations of photon-number distributions of various quantum optical fields and their coherence including photon-number correlations. These investigations have been based on the theoretical concepts and methods that had been established in the early years of quantum mechanics [16–19].
Twin beams (TWBs) generated in spontaneous parametric down-conversion and ideally composed of only photon pairs have been soon recognized as powerful and versatile sources of highly nonclassical light with nearly ideal photon pairing [4,19–23]. TWBs have directly been exploited for absolute detector calibration [5,24–26], quantum imaging [27] or sub-shot-noise imaging [28]. Considering more complex arrangements, photon-number-resolved post-selection on one beam of a TWB has allowed to generate sub-Poissonian states [12,21,29–33] useful, e.g., in sub-shot-noise measurements of absorption [28,34–37] or in increasing the capacity of communication channels [38]. PNRDs have also allowed to subtract multiple photons from TWBs to modify their properties including photon-number fluctuations [39] and correlations [13,33,40–42].
The simultaneous use of several TWBs with photon-number correlations interconnected via common photon-number-resolving measurements opens the door for new classes of states. The use of two TWBs and photon-number-resolved post-selection on their common signal beam has allowed to generate two-beam states with anti-correlations in photon-number fluctuations and marginal sub-Poissonian photon-number distributions [43] useful, e.g., in sub-shot-noise measurements of two-photon absorption coefficients. In general, this approach offers tailoring of complex properties of joint photon-number distributions including the nonclassical ones.
Here, we demonstrate the versatility and power of this approach by considering three TWBs and photon-number-resolved post-selection that, in its ideal form, provides two-dimensional (2D) photon-number distributions with the probabilities forming checkered patterns. Such states represent, in certain sense, 2D generalizations of the even and odd coherent states [19].
From the point of view of their statistical properties, the varying correlations of photon-number fluctuations represent the most important feature of these states: TWBs with ideal correlations in photon-number fluctuations and the states with ideal anti-correlations in photon-number fluctuations obtained by ideal photon-number-resolved post-selection lie at the borders of the investigated group of states. The varying statistical properties give application potential to these states. The bunching effect of the signal and idler photons in TWBs is used for efficient two-photon excitations, e.g., of markers in two-photon fluorescence microscopes [44]. On the other hand, the states with anti-correlations in photon-number fluctuations also exhibit sub-Poissonian marginal photon-number distributions that give them the ability to measure two-photon absorption coefficients with sub-shot-noise precision [43]. In general, the analyzed states exhibit both types of the discussed limiting behavior with the varying ratio which results in the varying sensitivity when monitoring two-photon absorption coefficients. This might be useful for two-photon spectroscopy that relies on the nonclassical light and its correlations [45,46]. Similarly, the properties of the analyzed states may be exploited for the preparation of specific atomic or molecular states via two-photon absorption.
The experiments involving several TWBs may be demanding for their geometry. However, the use of PNRDs with spatial multiplexing (cameras) allows in many cases simple and robust experimental implementations owing to their spatial resolution. Here, we demonstrate this advantage by considering two geometrically corresponding portions of the emission cone of the fields generated in type-I spontaneous parametric down-conversion [see Fig. 1(a)]. Inside them we define three entangled TWBs composed of their signal and idler beams, I$^\textrm {si}_{23}$, II$^\textrm {si}_{13}$ and III$^\textrm {si}_{12}$. Three optical beams under investigation are defined such that there exist comparable pairwise correlations among them: Beams 1, 2, and 3 are in turn composed of the beams II$^\textrm {s}_1$ and III$^\textrm {s}_1$, III$^\textrm {i}_2$ and I$^\textrm {i}_2$, and II$^\textrm {i}_3$ and I$^\textrm {s}_3$, as it is shown in Fig. 1(b). Then, we measure the number of photons in, say, beam 3 and analyze the properties of the remaining beams 1 and 2.
Fig. 1. (a) The signal- and idler-beam portions of the emission cone of spontaneous parametric down-conversion in which three TWBs are defined: I$^\textrm {si}_{23}$, II$^\textrm {si}_{13}$, III$^\textrm {si}_{12}$. (b) Scheme for constituting beams 1, 2 and 3 with pairwise photon-number correlations formed by the TWBs I$^\textrm {si}_{23}$, II$^\textrm {si}_{13}$ and III$^\textrm {si}_{12}$. (c) Nonzero probabilities of photon-number distribution $p_{12}(n_1,n_2;n_3)$ in the ideal case with $n_1 + n_2 = n_3 \equiv 1$, double arrow indicates ’adding’ photon pairs from TWB III$^\textrm {si}_{12}$ into the beams after ideal post-selection by detecting $n_3 = 1$ photon in beam 3.
To understand the origin of the checkered patterns in probabilities of photon-number distributions $p_{12}(n_1,n_2;n_3)$ of beams 1 and 2 after detecting $n_3$ photons in beam 3, we first omit the contribution of TWB III$^\textrm {si}_{12}$ to the distribution $p_{12}$. Ideal detection in beam 3 leaves nonzero only the probabilities $p_{12}(n_1,n_2;n_3)$ for which $n_1 + n_2 = n_3$. The checkered pattern of probabilities in the distribution $p_{12}$ then occurs after the inclusion (via the convolution) of the contribution of TWB III$^\textrm {si}_{12}$ with its photon-pair structure. The reason is that the convolution leaves nonzero only the probabilities of detecting $n_3$, $n_3 +2$, $n_3 + 4, \ldots$ overall photons in fields 1 and 2.
In the real experiment the checkered patterns in probabilities may be concealed by limited detection efficiency of the post-selecting detector, its dark counts and noises in the experimental TWBs. Nevertheless, interesting statistical properties of the generated fields with application potential still survive. Detailed analysis has shown that the most interesting statistical properties of such fields occur when the contribution of the post-selected beams with anti-correlation in photon-number fluctuations is comparable in weight to that of TWB III$^\textrm {si}_{12}$ with correlations in photon numbers.
We present our investigations of the studied states as follows. In Sec. 2, we introduce a multi-mode Gaussian model useful for theoretical investigations, describe the experimental setup and reconstruction of the experimental beams. Suitable quantities for characterizing the beams and their non-classicality are introduced in Sec. 3. Sec. 4 is devoted to the experimental characterization of the generated fields. For ideal photon-number-resolved post-selection, the investigated fields are analyzed in Sec. 5 relying on complete 3D reconstruction of the beams. Nonclassical properties of a typical joint photon-number distribution $p_{12}$ obtained by an iCCD camera are studied in Sec. 6. The checkered patterns in probabilities of the photon-number distributions $p_{12}$, are discussed in Sec. 7 considering ideal photon-number-resolving post-selection. Conclusions are drawn in Sec. 8. Fitting the Gaussian model to the experimental data is discussed in Appendix.
2. Gaussian theoretical model, experimental setup and field reconstruction
The analyzed two-beam fields are derived by photon-number-resolving post-selection from a Gaussian three-beam field composed of three TWBs such that it is endowed with bi-partite photon-number correlations. The joint photon-number distribution $p_{123}^{p}(n_1,n_2,n_3)$ of the three-beam field in its ideal noiseless form is expressed as
The Mandel-Rice photon(-pair) number distribution $p^{\textrm{MR}}$ used in Eq. (1),
characterizes a thermal $M$-mode field with $B$ mean photons (photon-pairs) per mode. In Eqs. (1) and (2), the Kronecker symbol $\delta$ and the gamma function $\Gamma$ are used. In Eq. (1), the mean photon-pair numbers $B_{12}$, $B_{23}$, and $B_{13}$ and the accompanying numbers $M_{12}$, $M_{23}$, and $M_{13}$ of modes refer in turn to TWBs III${}^\textrm {si}_{12}$, I${}^\textrm {si}_{23}$, and II${}^\textrm {si}_{13}$. The noises present in the three-beam field are modelled by additional multi-mode thermal fields whose photon-number distributions in the above Mandel-Rice form with $M_i$ modes and $B_i$ mean photon numbers per mode, $i=1,2,3$, are convolved with that of the noiseless field in Eq. (1):The analyzed fields are left in beams 1 and 2 provided that a given number $c_3$ of photocounts in beam 3 is registered by a PNRD. Their 2D photon-number distributions $p_{12}(n_1,n_2;c_3)$ are derived in the model as follows
using the detection matrix $T_3(c_3,n_3)$ of the PNRD present in beam 3. The elements of the detection matrix $T_3(c_3,n_3)$ give the probability of detecting $c_3$ photocounts out of $n_3$ photons illuminating the detector [see Eq. (19) in Appendix].The use of two additional PNRDs with their detection matrices $T_1$ and $T_2$ to monitor the studied 2D photon-number distributions $p_{12}(n_1,n_2;c_3)$ of the post-selected beams leaves us with the following 3D theoretical photocount distribution $f_{123}^\textrm {th}(c_1,c_2,c_3)$ giving the probability of simultaneous detection of $c_i$ photocounts in beam $i$ for $i=1,2,3$:
To verify the theoretical predictions for the fields properties we made the following experiment to arrive at the corresponding experimental photocount histogram $f_{123}(c_1,c_2,c_3)$. The original TWB whose signal and idler constituents were divided into the needed TWBs I$^\textrm {si}_{23}$, II$^\textrm {si}_{13}$, and III$^\textrm {si}_{12}$ according to the scheme in Figs. 1(a,b) was generated in a 5-mm-long type-I beta-barium-borate crystal (BaB${}_2$O${}_4$, BBO) cut for a slightly non-collinear geometry. The experimental setup as well as the division of the signal and idler detection strips into TWBs I$^\textrm {si}_{23}$, II$^\textrm {si}_{13}$, and III$^\textrm {si}_{12}$ are shown in Fig. 2. Spontaneous parametric down-conversion as the source of the original TWB was pumped by the third-harmonic pulses (280 nm) generated by pulses of a femtosecond cavity dumped Ti:sapphire laser (840 nm, 150 fs). The nearly-frequency-degenerate signal and idler photons ($\approx$560 nm) were first filtered by a 14-nm-wide bandpass interference filter and then detected in different strips of the photocathode of iCCD camera Andor DH334-18U-63. The pump beam was actively stabilized via a motorized half-wave plate followed by a polarizer and a detector that gave information about the actual pump intensity. The errors of the determined quantities were derived from the number $1.2\times 10^6$ of experimental repetitions following the usual rules of error propagation.
Fig. 2. (a) Experimental setup: A TWB is generated in a nonlinear crystal BBO by ultra-short third-harmonic pulses whose intensity is actively stabilized by a feedback provided by rotating half-wave plate HWP, polarizing beam splitter PBS, and detector D. The signal and the idler beams (after reflection on mirror HR) are filtered by a bandpass interference filter F and then detected by an iCCD camera. (b) 3D photocount histogram $f_{123}$ is extracted from the experimental data acquired in three indicated detection areas 1, 2, and 3 defined inside the signal and idler detection strips, also the noise is monitored in strip N.
We note that the used simple experimental scheme with post-selection from the obtained experimental data does not have to be suitable for investigating samples prone to photo-bleaching. In this case, a suitable feed-forward scheme has to be applied to arrive at the post-selected contribution of the analyzed fields (e.g., by combining the setups from [43] and [37]) that is subsequently incoherently superimposed on a usual TWB.
We have applied the method of absolute detector calibration [25] to arrive at the values of detection efficiencies $\eta$: $\eta _1 = 0.227\pm 0.005$, $\eta _2 = 0.234\pm 0.005$, $\eta _3 = 0.231\pm 0.005$. Then, requiring coincidence of the first and the second theoretical and experimental (integrated) intensity moments we have minimized the declination between the experimental photocount histogram $f_{123}$ and the theoretical photocount distribution $f_{123}^\textrm {th}$ to obtain suitable values of the model parameters: $B_{12} = 0.071\pm 0.010$, $M_{12} = 40\pm 5$, $B_{23} = 0.034\pm 0.005$, $M_{23} = 75 \pm 10$, $B_{13} = 0.042\pm 0.005$, $M_{13} = 61\pm 10$, $B_{1} = 5.4\pm 0.5$, $M_{1} = 0.015\pm 0.005$, $B_{2} = 6.5\pm 0.5$, $M_{2} = 0.008\pm 0.003$, $B_{3} = 5.8\pm 0.5$, $M_{3} = 0.018\pm 0.005$. Details are given in Appendix. We note that the noise distributions with numbers $M_{1}$, $M_{2}$ and $M_{3}$ of modes considerably lower than 1 are highly peaked at the value $n = 0$, which is a consequence of the specific form of the noise occurring in the detection process specific to an iCCD camera (electronic/readout noise, software identification of photocount). According to the values of the fitted parameters, the overall three-beam theoretical Gaussian field is composed of two TWBs with 2.55 ($M_{13}B_{13}$) and 2.57 ($M_{23}B_{23}$) mean photon pairs that participate in post-selection. The last TWB with 2.83 ($M_{12}B_{12}$) mean photon pairs directly contributes to the statistics of the analyzed photon-number distributions $p_{12}(n_1,n_2;c_3)$, together with the single-beam noise fields of mean photon numbers 0.08 ($M_1B_1$)and 0.05 ($M_2B_2$). The single-beam noise field with 0.1 mean photons ($M_3B_3$) affects the detection used for post-selection. Each beam was detected by $N_1 = N_2 = N_3 = 3024$ macropixels (composed of hardware-binned $8 \times 8$ pixels of the iCCD chip) that gave $d_1 = d_2 = d_3 = 0.15 \pm 0.02$ mean noise counts per detection window and detection area.
3. Properties of the post-selected fields and their quantification, non-classicality
The properties of the analyzed two-beam fields generated by post-selection depend on the post-selecting number $c_3$ of photocounts detected in beam 3. Whereas the post-selected contribution of the two-beam fields [arising in TWBs II$^\textrm {si}_{13}$ and I$^\textrm {si}_{23}$, see Fig. 1(b)] exhibits anti-correlations in photon-number fluctuations, the remaining contribution built by TWB III$^\textrm {si}_{12}$ is endowed with correlations in photon-number fluctuations. To quantify their relative influence, we determine covariance $C_{n,\Delta }$ ($C_{c,\Delta }$) of the photon-number (photocount) fluctuations $\Delta n$ ($\Delta c$) along the formula:
Both contributions may be responsible for nonclassical photon-number correlations identified by the values of the (modified) noise-reduction-parameter $R_{n,-}$ ($R_{n,+}$) smaller than 1:
We note that $R_{n,\pm } = 1$ for two independent Poissonian fields in coherent states.
Post-selection also significantly affects the relative widths of the marginal photon-number distributions of beams 1 and 2. They are quantified by the corresponding Fano factors $F_{n,j}$,
Non-classicality of the post-selected fields can be evidenced by suitable non-classicality criteria (NCCa). Detailed analysis of the NCCa for 2D photon-number distributions presented in [47] identified the NCC $C_W$ as the most powerful for our type of nonclassical states:
The moments of integrated intensity $W$ occurring in Eq. (9) refer to the normally-ordered photon-number moments that arise in the detection theory [18]. For the relation between the integrated-intensity moments and the usual photon-number moments, see Eq. (16) in Appendix.
The NCC $C_W$ in Eq. (9) belongs to a whole group of the NCCa $C$ originating in the Cauchy–Schwarz inequality [48]:
The Mandel detection formula [18,19] allows to map [49,50] this group of NCCa in intensity moments to the corresponding group of NCCa written in probabilities $p(k_1,k_2)$ of simultaneously detecting $k_1$ photons in mode 1 and $k_2$ photons in mode 2 [47]:
We may get useful information about the location of non-classicality across the profile of photon-number distribution $p(k_1,k_2)$ provided that we put suitable constraints to the varying indices $k_j$ and $l_j$, $j=1,2$, of the NCCa $\bar {C}_{k_1 k_2}^{l_1 l_2}$ for probabilities that emerge from the NCCa $C$ in Eq. (10) via the mapping (11). This leads us to the following NCCa:
The Lee non-classicality depth (NCD) [51] may be assigned to both types of the above NCCa to quantify the non-classicality. To do this, we first have to determine the value $s_\textrm {th}$ of the field ordering parameter at which the considered NCC is no more able to reveal the non-classicality. We note that any nonclassical field fully exhibits its nonclassical properties in the normal field ordering $s=1$. When we consider a general field ordering with the given parameter $s$, a noise thermal field with mean photon number $(1-s)/2$ is effectively superimposed on the original normally-ordered field. This results in (partial) loss of the nonclassical properties. The greater the noise mean photon number $(1-s_\textrm {th})/2$ at which the non-classicality is lost is, i.e. the smaller the threshold value $s_\textrm {th}$ is, the more nonclassical the field is (for more details, see [18,48,52]). This brings us to the definition of NCD $\tau$:
According to Eq. (13) $\tau = 0$ for any classical field and in general $\tau \le 1$.
Provided that the ordering parameter $s$ is greater than the threshold value $s_\textrm {th}$, the quasi-distribution $P_{12,s}(W_1,W_2)$ of integrated intensities determined for this ordering attains negative values [16,17] that represent a direct evidence of the non-classicality. The quasi-distribution $P_{12,s}(W_1,W_2)$ is derived from the corresponding photon-number distribution $p_{12}(n_1,n_2)$ by applying the following decomposition into the Laguerre polynomials $L_k$ [18,53]:
4. Nonclassical two-beam fields of varying intensity post-selected by an iCCD camera
First we analyze the two-beam fields present in the experimental setup in which post-selection is performed by an iCCD camera with detection efficiency around 23 %. The states post-selected by different photocount numbers $c_3$ in beam 3 differ in relative weights of two constituting contributions. Whereas the absolute weight of the twin-beam contribution (TWB III$^\textrm {si}_{12}$ in the model) with photon-number correlations remains the same for all post-selected fields, the absolute weights of the post-selected contributions (post-selected TWBs II$^\textrm {si}_{13}$ and I$^\textrm {si}_{23}$ in the model) with anti-correlations in photon-number fluctuations naturally increase with the increasing post-selecting photocount number $c_3$ up to $c_3 = 7$ [see Fig. 3(a)] where dark counts in the detection of beam 3 start to play a qualitative role in the post-selection process.
Fig. 3. (a) Mean number of photons $\langle n_1\rangle$ (photocounts $\langle c_1\rangle$) and (b) Fano factor $F_{n_1}$ ($F_{c_1}$) of beam 1, (c) covariance $C_{n,\Delta }$ ($C_{c,\Delta }$), (d) noise-reduction-parameter $R_{n,-}$ ($R_{c,-}$), (e) modified noise-reduction-parameter $R_{n,+}$ ($R_{c,+}$), and (f) non-classicality depth $\tau _{C_W}$ of the two-beam fields as they depend on the post-selecting photocount number $c_3$ of beam 3. Isolated symbols are drawn for the experimental photocount histograms (red $\color{red}{\ast}$) and fields reconstructed by 2D maximum-likelihood approach (green $\color{green}{\triangle}$); solid blue curves originate in 3D Gaussian model. The horizontal dashed lines indicate the borders of non-classicality regions ($F = 1$, $R_{-} = 1$).
In Fig. 3, we plot the quantities characterizing the experimental photocount histograms (red $\ast$) side-by-side with those belonging to the reconstructed two-beam fields reached by 2D maximum-likelihood approach (green ${\triangle}$, for details, see [6,54]) and 3D Gaussian fit (plain blue curves, for details, see Appendix). We note that both reconstruction methods correct the experimental data for limited detection efficiencies, dark-count rates and finite numbers of detection macro-pixels of PNRDs in beams 1 and 2. The Gaussian fit reveals that the constant twin-beam contribution consists of 2.83 mean photon pairs whereas the post-selected contributions vary their mean photon numbers from around 2 to 4 in each beam as the post-selecting photocount number $c_3$ increases from 0 to 7.
The marginal Fano factors $F$ of the twin-beam contribution are greater than 1 owing to the super-Poissonian character of spontaneous parametric down-conversion. It was shown in [43] that the marginal Fano factors $F$ of the post-selected contribution decrease with the increasing post-selecting photocount number $c_3$ and sub-Poissonian fields can be reached this way (see the next Sec. 5). We observe such behavior in Fig. 3(b) where the increasing weight of the post-selected contribution allows to reach the marginal Fano factors $F \approx 1$ for $c_3 \in \langle 3,5 \rangle$. We note that for $c_3 = 3$ the weights of both contributions become comparable and the post-selected contribution prevails for $c_3 > 3$.
Strong correlations of photon-number fluctuations characterize the twin-beam contribution composed of photon pairs. On the other hand, the post-selected contribution is endowed with anti-correlations in photon-number fluctuations that gradually weaken the photon-number correlations of the post-selected fields as the post-selecting photocount number $c_3$ increases. This behavior is quantified at the classical level in Fig. 3(c) where the covariances $C_{c,\Delta }$ and $C_{n,\Delta }$ of the photocount and photon-number fluctuations $\Delta c$ and $\Delta n$ of beams 1 and 2 are drawn as they depend on the photocount number $c_3$. In general, both contributions may be responsible for non-classicality of the post-selected fields. The values of the noise-reduction-parameters $R_{c,-}$ and $R_{n,-}$ plotted in Fig. 3(d), though increasing with the increasing photocount number $c_3$, remain below the classical limit 1. This reflects the fact that the twin-beam component with photon-number correlations assures the non-classicality of the fields despite its decreasing relative weight. On the other hand, the values of the modified noise-reduction-parameters $R_{c,+}$ and $R_{n,+}$ shown in Fig. 3(e), though decreasing with the increasing photocount number $c_3$, are above 1, i.e. in the classical regime. This means that in the analyzed fields anti-correlations in photon-number fluctuations of the post-selected contribution are concealed by the correlations of the twin-beam contribution.
However, the increasing weight of anti-correlations of photon-number fluctuations of the post-selected contribution weakens the overall non-classicality of the generated two-beam fields, as evidenced by the NCD $\tau _{C_W}$ belonging to the NCC $C_W$ and plotted in Fig. 3(f).
5. Nonclassical two-beam fields of varying intensity post-selected by an ideal detector
To reveal in certain sense ideal (optimal) properties of the analyzed two-beam fields, we discuss the behavior of these fields attained by post-selection by an ideal PNRD (with unit detection efficiency and no dark counts). Such ideal detector would provide the post-selected contribution with much stronger anti-correlations in photon-number fluctuations than that obtained by the used iCCD camera. This considerably increases the non-classicality of the obtained states. Whereas the weight of the twin-beam contribution remains the same (2.83 mean photon pairs), the post-selected contribution increases the mean photon numbers from around 0 to 6 as the post-selecting photon number $n_3$ of beam 3 increases [see Fig. 4(a)]. The ideal post-selection gives the marginal sub-Poissonian fields in beams 1 and 2 with the Fano factors $F_n^{\textrm{id}} < 1$ for the post-selecting numbers $n_3 \ge 2$ [see Fig. 4(b)]. We note that for $n_3 = 3$ the weight of the post-selected contribution is only about one half of the twin-beam contribution. For $n_3 \in \langle 2,12\rangle$, the post-selected fields are sub-Poissonian with the increasing weight of the post-selected contribution and thus also increasing mean photon numbers as the post-selecting number $n_3$ increases. The noise present in the original three-beam field begins to seriously degrade the post-selection process for $n_3 > 10$ as it is apparent from the Fano factors $F_n^{\textrm{id}}$ plotted in Fig. 4(b).
Fig. 4. (a) Mean number of photons $\langle n_1\rangle^{\textrm{id}}$ and (b) Fano factor $F_{n_1}^{\textrm{id}}$ of beam 1, (c) covariance $C_{n,\Delta }^{\textrm{id}}$, (d) noise-reduction-parameter $R_{n,-}^{\textrm{id}}$, (e) modified noise-reduction-parameter $R_{n,+}^{\textrm{id}}$, and (f) non-classicality depth $\tau _{C_W}^{\textrm{id}}$ of the two-beam fields as they depend on the post-selecting photon number $n_3$ of beam 3. Isolated symbols are drawn for the reconstructed fields reached by 3D maximum-likelihood approach (dark green $\diamond$); solid dark blue curves originate in 3D Gaussian model. The horizontal dashed lines indicate the borders of non-classicality regions ($F = 1$, $R_{\pm } = 1$).
The increasing weight of the post-selected contribution considerably lowers the original correlations in photon-number fluctuations of the twin-beam contribution as evidenced in Fig. 4(c) where the covariance $C_{n,\Delta }^{\textrm{id}}$ of photon-number fluctuations is plotted. Whereas the covariance $C_{n,\Delta }^{\textrm{id}} \approx 0.7$ for the only twin-beam contribution, it reaches values around 0.2 in the area with the strongest influence of the post-selected contribution occurring for $n_3 \in \langle 7,12\rangle$. In this area of photon numbers $n_3$, the values of noise-reduction-parameter $R_{n,-}^{\textrm{id}} \approx 0.7$ indicate the non-classicality originating in photon pairing at the level comparable to that reached by post-selection by the iCCD camera. On the other hand, the values of modified noise-reduction-parameter $R_{n,+}^{\textrm{id}}$ lie around 1 which is the border of non-classicality. It contrasts with the lowest attainable values $R_{n,+} \approx 1.3$ reached by the iCCD camera. This improvement in the values of the modified noise-reduction-parameter $R_{n,+}$ is caused by two reasons: much stronger anti-correlations in photon-number fluctuations and greater relative weights of the post-selected contribution for the ideal detector. It is worth noting that the 3D Gaussian model with its curves plotted in Figs. 4(d,e) indicates the existence of nonclassical states obeying simultaneously the non-classicality inequalities $R_{n,-} < 1$ and $R_{n,+}< 1$ based on, in certain sense, complementary quantities $n_1 - n_2$ and $n_1 + n_2$ (for $n_3 \in \langle 8,11\rangle$).
Similarly to when the iCCD camera is used for post-selection, the generated two-beam states are identified as nonclassical by the NCC $C_W$ [see Fig. 4(f)]. Whereas the two-beam fields post-selected by the ideal detector with small post-selecting numbers $n_3$ are more nonclassical than their counterparts reached by the iCCD camera and small post-selecting photocount numbers $c_3$ ($\tau \approx 0.3$ vs. $\tau \approx 0.22$), the opposed is true for greater photon (photocount numbers) $n_3$ ($c_3$) for which the modified noise-reduction-parameters $R_{n,+}$ attain their lowest values ($\tau \approx 0.1$ vs. $\tau \approx 0.16$). This reflects the complexity of the non-classicality quantified by the NCD $\tau$.
6. Typical properties of two-beam fields reached by an iCCD camera
We address in detail the properties of typical experimental two-beam fields reached by the iCCD camera. As an example, we analyze the properties of the field obtained by registering $c_3 =3$ photocounts. In this field, both the twin-beam and the post-selected contributions are balanced. As apparent from its photon-number distribution $p_{12}(n_1,n_2)$ shown in Fig. 5(a), the post-selected contribution considerably broadens the photon-number distribution of the twin-beam contribution in the form of a narrow cigar lying around the diagonal $n_1 = n_2$. The corresponding quasi-distribution $P_{12}(W_1,W_2)$ of integrated intensities $W_1$ and $W_2$ drawn in Fig. 5(b) even attains the form of a blurred cross with one arm along the diagonal $n_1 = n_2$, i.e. it recognizes both contributions. Though the quasi-distribution $P_{12}$ of integrated intensities is plotted for the symmetric ordering $s=0$, i.e. where the non-classicality is already weak, it contains larger areas with slightly negative values drawn in black in Fig. 5(b).
Fig. 5. (a) Photon-number distribution $p_\textrm {12}(n_1,n_2)$ and (b) the corresponding quasi-distribution $P_{12,s=0}(W_1,W_2)$ of integrated intensities for $c_3 = 3$ post-selecting photocounts in beam 3. (c) Non-classicality depth $\bar {\tau }_{\bar {C}_p}$ as a function of photon numbers $n_1$ and $n_2$ in beams 1 and 2 considering only the NCCa with the averaged probability greater than 0.005 for $c_3 = 3$.
Whereas the NCC $C_W$ based on intensity moments assigns the NCD $\tau _{C_W} \approx 0.18$ to this field [see Fig. 3(f)], the NCCa $\bar {C}_p$ that use the probabilities of the photon-number distribution $p_{12}$ provide the NCDs $\bar {\tau }_{\bar {C}_p}$ up to 0.35, as documented in Fig. 5(c). This accords with the general conclusion that the NCCa based on probabilities are more efficient in identifying and quantifying the non-classicality [43,48].
7. Typical properties of two-beam fields reached by an ideal detector
The checkered patterns in probabilities of photon-number distributions $p_{12}$ of the two-beam fields arising in high-quality post-selection by an ideal PNRD are the most striking feature of these fields. The checkered pattern shown in Fig. 6(a) for the field reconstructed by 3D maximum-likelihood approach [43] and post-selected by photon number $n_3 = 9$ originates in the convolution of two 2D photon-number distributions with highly developed photon-number correlations (twin-beam contribution) and highly developed anti-correlations in photon-number fluctuations (post-selected contribution) [see the scheme in Fig. 1(c)]. The quasi-distribution $P_{12}$ of integrated intensities drawn in Fig. 6(b) for $s=0.1$ does not already allow to distinguish the checkered pattern as this is concealed by the detection noise stemming from the used field-operator ordering. Nevertheless, its form of the blurred rotated cross clearly identifies both contributions, similarly as in the case of the quasi-distribution $P_{12}$ in Fig. 5(b) obtained with the iCCD camera. The non-classicality is still strongly present for $s=0.1$ and it forms crescent areas with negative values in the vicinity of the $W_1$ and $W_2$ axes.
Fig. 6. (a) Photon-number distribution $p_\textrm {12}^{\textrm{id}}(n_1,n_2)$ and (b) the corresponding quasi-distribution $P_{12,s=0.1}^{\textrm{id}}(W_1,W_2)$ of integrated intensities for $n_3 = 9$ post-selecting photons in beam 3; (c) [(d)] photon-number distribution $p_\textrm {12}^{\textrm{id}}(n_1,n_2)$ for $n_3 = 8$ [10]. (e) Non-classicality depths $\bar {\tau }_{\bar {C}_p}^{\textrm{id}}$ of the NCCa $\bar {C}_p$ as they depend on photon numbers $n_1$ and $n_2$ in beams 1 and 2 considering only the NCCa with the averaged probability greater than 0.005 and $n_3 = 9$.
We may have a look at the photon-number distribution $p_{123}$ of the whole three-beam field reconstructed by 3D maximum likelihood approach. We recover a 3D checkered pattern in probabilities. The cut of $p_{123}(n_1,n_2,n_3)$ for $n_3 = 9$ plotted in Fig. 6(a) emphasizes the probabilities with odd sum $n_1 + n_2$. On the other hand, the probabilities with even sums $n_1 + n_2$ prevail in the neighbor cuts $p_{123}(n_1,n_2,n_3)$ for $n_3 = 8$ and 10 shown in Figs. 6(c,d). In general, the cuts by even (odd) photon numbers $n_3$ prefer the probabilities with even (odd) sums $n_1 + n_2$. This is a natural consequence of the fact that the three-beam field is mainly composed of three types of photon pairs.
The NCC $C_W$ based on intensity moments quantifies the non-classicality by the NCD $\tau _{C_W}^{\textrm{id}} \approx 0.12$ [see Fig. 4(f)]. In contrast, its counterpart with probabilities provides the maxima of NCDs $\bar {\tau }_{\bar {C}_p} \approx 0.45$, as documented in Fig. 6(e).
Direct generation of the states with checkered patterns in photon-number distributions is challenging. Nevertheless, it would require a high quality post-selecting PNRD. According to the graphs in Fig. 7, detection efficiencies better than 75 % and a PNRD with low level of dark counts comparable to the used iCCD camera would be needed to directly generate these states.
Fig. 7. Threshold detection efficiencies $\eta _3^\textrm {th}$ of the post-selecting PNRD that allow the observation of photon-number distributions $p_{12}(n_1,n_2;c_3)$ with the sum of the prevailing probabilities (for either even or odd sum $n_1 + n_2$) exceeding the values (a) 0.55 and (b) 0.65 as they depend on the post-selecting photocount number $c_3$ of beam 3. The three-beam theoretical Gaussian field arising from fitting the experimental data is used; noisy ($d_3=0.15$, blue solid curves) as well as noiseless ($d_3=0$, black dashed curves) post-selecting PNRDs with $N_3 = 3024$ macropixels are considered.
8. Conclusions
We have generated a group of two-beam non-classical fields endowed with two distinct features of varying intensity. These are pairing of photons (leading to photon-number correlations) and anti-correlations in photon-number fluctuations originating in post-selection by a photon-number-resolving detector. These fields were generated by post-selection on one beam of a three-beam field constituted by three twin beams endowed with strong photon-number correlations. A simple experimental geometry using one emission cone of spontaneous parametric down-conversion in a crystal was applied owing to the use of spatial resolution of a photon-number-resolving iCCD camera.
The checkered patterns in probabilities of photon-number distributions are the most remarkable feature of these fields. They, however, need high quality of the photon-number-resolved post-selection to be generated. Though the post-selection provided by the iCCD camera in the performed experiment did not suffice to observe the checkered patterns directly in the experimental data, we have obtained a group of highly nonclassical states. The photon-number distributions with checkered patterns have then been demonstrated using the maximum-likelihood reconstruction that encompassed also the measurement on the post-selecting photon-number-resolving detector. Non-classicality of the generated fields has been certified by the non-classicality depths determined for suitable non-classicality quantifiers written both in intensity moments and probabilities of photon-number distributions.
The generated states are potentially useful for applications. The varying ratio of the present correlations and anti-correlations of photon-number fluctuations represents their unique feature that makes them attractive for two-photon excitations. Whereas correlations in photon-number fluctuations lead to photon bunching useful in efficient two-photon excitations (two-photon fluorescence microscopes), anti-correlations in photon-number fluctuations are accompanied by marginal fields sub-Poissonian photon-number statistics that allow to measure two-photon absorption coefficients with sub-shot-noise precision. The application of states with different ratios of correlations and anti-correlations and thus different susceptibility to two-photon absorption might be useful for two-photon spectroscopy with nonclassical light. Also the preparation of specific atomic and molecular states via two-photon absorption of these states is foreseen.
A. Appendix: Three-beam Gaussian fields, their moments and parameters
The three-beam Gaussian fields with their joint photon-number distribution $p_{123}(n_1,n_2,n_3)$ given in Eq. (3) can alternatively be characterized by the corresponding photon-number moments $\langle n_1^{k_1} n_2^{k_2} n_3^{k_3} \rangle$ defined as:
The statistics of photons inside the analyzed beams (as observed by ideal detection) are characterized by the moments $\langle W_1^{k_1} W_2^{k_2} W_3^{k_3} \rangle$ of the (integrated-) intensities defined as the normally-ordered photon-number moments [18]. The canonical commutation relations give the following relations between both types of moments:
In the experiment, we similarly define the photocount moments $\langle c_1^{k_1} c_2^{k_2} c_3^{k_3} \rangle$ using the experimental photocount histogram $f_{123}(c_1,c_2,c_3)$:
Also in this case, we may assign the intensity moments $\langle {\cal W}_1^{k_1} {\cal W}_2^{k_2} {\cal W}_3^{k_3} \rangle _E$ to the photocount moments $\langle c_1^{l_1} c_2^{l_2} c_3^{l_3} \rangle$ using the relations described in Eq. (16). The photocount moments $\langle c_1^{l_1} c_2^{l_2} c_3^{l_3} \rangle$ and the accompanying intensity moments $\langle {\cal W}_1^{k_1} {\cal W}_2^{k_2} {\cal W}_3^{k_3} \rangle _E$ as the experimental characteristics of the beams naturally play an important role in fitting the parameters of the constituting Gaussian beams.
Both types of the moments are mutually related via the properties of the used PNRDs. In case of iCCD cameras, the corresponding detection matrix can be expressed in the form [6]
The Gaussian three-beam field is characterized by 12 parameters: 3 numbers of modes of the constituting TWBs ($M_{12}$, $M_{23}$, $M_{13}$) with the accompanying 3 mean photon-pair numbers ($B_{12}$, $B_{23}$, $B_{13}$) and 3 numbers of modes of the noise beams ($M_j$, $j=1,2,3$) with their 3 mean photon numbers ($B_j$, $j=1,2,3$). On the other hand, we have at disposal 9 first- and second-order intensity moments belonging to photocounts: $\langle {\cal W}_j\rangle _E$, $\langle (\Delta {\cal W}_j)^2\rangle _E$, $j=1,2,3$ and $\langle \Delta {\cal W}_j \Delta {\cal W}_k \rangle _E$, $(j,k) = (1,2), (2,3), (1,3)$. Using the known detection efficiencies $\eta _j$, $j=1,2,3$, the following two sets of relations are established between these experimental intensity moments and the theoretical intensity moments of the fields constituting the three-beam field:
In finding the minimum of the declination function ${\cal D}$, we may vary the second-order intensity moments $\langle (\Delta W_{jk})^2\rangle$, $(j,k)= (1,2), (2,3), (1,3)$, whose values are naturally bounded from both sides:
Funding
Grantová Agentura České Republiky (18-08874S, J.P., O.H., R.M., V.M).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003). [CrossRef]
2. O. Haderka, M. Hamar, and J. Peřina Jr., “Experimental multi-photon-resolving detector using a single avalanche photodiode,” Eur. Phys. J. D 28(1), 149–154 (2004). [CrossRef]
3. I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-Gaussian multiphoton light,” npj Quantum Inf. 4(1), 4 (2018). [CrossRef]
4. O. Haderka, J. Peřina Jr., M. Hamar, and J. Peřina, “Direct measurement and reconstruction of nonclassical features of twin beams generated in spontaneous parametric down-conversion,” Phys. Rev. A 71(3), 033815 (2005). [CrossRef]
5. G. Brida, I. P. Degiovanni, M. Genovese, M. L. Rastello, and I. R. Berchera, “Detection of multimode spatial correlation in PDC and application to the absolute calibration of a CCD camera,” Opt. Express 18(20), 20572–20584 (2010). [CrossRef]
6. J. Peřina Jr., M. Hamar, V. Michálek, and O. Haderka, “Photon-number distributions of twin beams generated in spontaneous parametric down-conversion and measured by an intensified CCD camera,” Phys. Rev. A 85(2), 023816 (2012). [CrossRef]
7. D. S. Tasca, R. S. Aspden, P. A. Morris, G. Anderson, R. W. Boyd, and M. J. Padgett, “The influence of non-imaging detector design on heralded ghost-imaging and ghost-diffraction examined using a triggered iCCD camera,” Opt. Express 21(25), 30460 (2013). [CrossRef]
8. R. Chrapkiewicz, W. Wasilewski, and K. Banaszek, “High-fidelity spatially resolved multiphoton counting for quantum imaging applications,” Opt. Lett. 39(17), 5090–5093 (2014). [CrossRef]
9. A. Avella, I. Ruo-Berchera, I. P. Degiovanni, G. Brida, and M. Genovese, “Absolute calibration of an EMCCD camera by quantum correlation linking photon counting to analog regime,” Opt. Lett. 41(8), 1841–1844 (2016). [CrossRef]
10. L. Qi, F. Just, G. Leuchs, and M. V. Chekhova, “Autonomous absolute calibration of an iCCD camera in single-photon detection regime,” Opt. Express 24(23), 26444–26453 (2016). [CrossRef]
11. A. J. Miller, S. W. Nam, J. M. Martinis, and A. V. Sergienko, “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. 83(4), 791–793 (2003). [CrossRef]
12. G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116(14), 143601 (2016). [CrossRef]
13. O. S. Magańa-Loaiza, R. de J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019). [CrossRef]
14. M. Ramilli, A. Allevi, V. Chmill, M. Bondani, M. Caccia, and A. Andreoni, “Photon-number statistics with silicon photomultipliers,” J. Opt. Soc. Am. B 27(5), 852–862 (2010). [CrossRef]
15. G. Chesi, L. Malinverno, A. Allevi, R. Santoro, M. Caccia, and M. Bondani, “Measuring nonclassicality with silicon photomultipliers,” Opt. Lett. 44(6), 1371–1374 (2019). [CrossRef]
16. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131(6), 2766–2788 (1963). [CrossRef]
17. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10(7), 277–279 (1963). [CrossRef]
18. J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena (Kluwer, 1991).
19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
20. O. Jedrkiewicz, Y. K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. A. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down-conversion,” Phys. Rev. Lett. 93(24), 243601 (2004). [CrossRef]
21. M. Bondani, A. Allevi, G. Zambra, M. G. A. Paris, and A. Andreoni, “Sub-shot-noise photon-number correlation in a mesoscopic twin beam of light,” Phys. Rev. A 76(1), 013833 (2007). [CrossRef]
22. J.-L. Blanchet, F. Devaux, L. Furfaro, and E. Lantz, “Measurement of sub-shot-noise correlations of spatial fluctuations in the photon-counting regime,” Phys. Rev. Lett. 101(23), 233604 (2008). [CrossRef]
23. G. Brida, L. Caspani, A. Gatti, M. Genovese, A. Meda, and I. R. Berchera, “Measurement of sub-shot-noise spatial correlations without backround subtraction,” Phys. Rev. Lett. 102(21), 213602 (2009). [CrossRef]
24. D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelectric detectors,” Sov. J. Quantum Electron. 10(9), 1112–1117 (1980). [CrossRef]
25. J. Peřina Jr., O. Haderka, M. Hamar, and V. Michálek, “Absolute detector calibration using twin beams,” Opt. Lett. 37(13), 2475–2477 (2012). [CrossRef]
26. O. Haderka, J. Peřina Jr., V. Michálek, and M. Hamar, “Absolute spectral calibration of an intensified CCD camera using twin beams,” J. Opt. Soc. Am. B 31(10), B1–B7 (2014). [CrossRef]
27. M. Genovese, “Real applications of quantum imaging,” J. Opt. 18(7), 073002 (2016). [CrossRef]
28. G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4(4), 227–230 (2010). [CrossRef]
29. J. Rarity and P. Tapster, “Quantum interference: experiments and applications,” Philos. Trans. R. Soc., A 355(1733), 2267–2277 (1997). [CrossRef]
30. J. Laurat, T. Coudreau, N. Treps, A. Maitre, and C. Fabre, “Conditional preparation of a quantum state in the continuous variable regime: Generation of a sub-Poissonian state from twin beams,” Phys. Rev. Lett. 91(21), 213601 (2003). [CrossRef]
31. J. Peřina Jr., O. Haderka, and V. Michálek, “Sub-Poissonian-light generation by postselection from twin beams,” Opt. Express 21(16), 19387–19394 (2013). [CrossRef]
32. M. Lamperti, A. Allevi, M. Bondani, R. Machulka, V. Michálek, O. Haderka, and J. Peřina Jr., “Optimal sub-Poissonian light generation from twin beams by photon-number resolving detectors,” J. Opt. Soc. Am. B 31(1), 20–25 (2014). [CrossRef]
33. T. S. Iskhakov, V. C. Usenko, U. L. Andersen, R. Filip, M. V. Chekhova, and G. Leuchs, “Heralded source of bright multi-mode mesoscopic sub-Poissonian light,” Opt. Lett. 41(10), 2149–2152 (2016). [CrossRef]
34. E. Jakeman and J. G. Rarity, “The use of pair production processes to reduce quantum noise in transmission measurements,” Opt. Commun. 59(3), 219–223 (1986). [CrossRef]
35. R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Absorption spectroscopy at the ultimate quantum limit from single-photon states,” New J. Phys. 19(2), 023013 (2017). [CrossRef]
36. M. Li, C.-L. Zou, D. Liu, G.-P. Guo, G.-C. Guo, and X.-F. Ren, “Enhanced absorption microscopy with correlated photon pairs,” Phys. Rev. A 98(1), 012121 (2018). [CrossRef]
37. J. Sabines-Chesterkind, A. R. McMillan, P. A. Moreau, S. K. Josh, S. Knauer, E. Johnston, J. G. Rarity, and J. C. F. Matthews, “Twin-beam sub-shot-noise raster-scanning microscope,” Opt. Express 27(21), 30810–30818 (2019). [CrossRef]
38. B. E. A. Saleh and M. C. Teich, “Can the channel capacity of a light-wave communication system be increased by the use of photon-number-squeezed light?” Phys. Rev. Lett. 58(25), 2656–2659 (1987). [CrossRef]
39. T. S. Iskhakov, V. C. Usenko, R. Filip, M. V. Chekhova, and G. Leuchs, “Low-noise macroscopic twin beams,” Phys. Rev. A 93(4), 043849 (2016). [CrossRef]
40. G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics,” Phys. Rev. A 46(1), 485–488 (1992). [CrossRef]
41. M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71(4), 043805 (2005). [CrossRef]
42. S. M. Barnett, G. Ferenczi, C. R. Gilson, and F. C. Speirits, “Statistics of photon subtracted and photon-added states,” Phys. Rev. A 98(1), 013809 (2018). [CrossRef]
43. J. Peřina Jr., V. Michálek, R. Machulka, and O. Haderka, “Two-beam light with simultaneous anti-correlations in photon-number fluctuations and sub-Poissonian statistics,” Phys. Rev. A 104(1), 013712 (2021). [CrossRef]
44. O. Varnavski and T. Goodson III, “Two-photon fluorescence microscopy at extremely low excitation intensity: The power of quantum correlations,” J. Am. Chem. Soc. 142(30), 12966–12975 (2020). [CrossRef]
45. B. E. A. Saleh, B. M. Jost, H.-B. Fei, and M. C. Teich, “Entangled-photon virtual-state spectroscopy,” Phys. Rev. Lett. 80(16), 3483–3486 (1998). [CrossRef]
46. J. Svozilík, J. Peřina Jr., and R. de León-Montiel, “Virtual-state spectroscopy with frequency tailored intense entangled beams,” J. Opt. Soc. Am. B 35(2), 460–467 (2018). [CrossRef]
47. J. Peřina Jr., I. I. Arkhipov, V. Michálek, and O. Haderka, “Non-classicality and entanglement criteria for bipartite optical fields characterized by quadratic detectors,” Phys. Rev. A 96(4), 043845 (2017). [CrossRef]
48. J. Peřina Jr., O. Haderka, and V. Michálek, “Non-classicality and entanglement criteria for bipartite optical fields characterized by quadratic detectors II: Criteria based on probabilities,” Phys. Rev. A 102(4), 043713 (2020). [CrossRef]
49. D. N. Klyshko, “Observable signs of nonclassical light,” Phys. Lett. A 213(1-2), 7–15 (1996). [CrossRef]
50. E. Waks, B. C. Sanders, E. Diamanti, and Y. Yamamoto, “Highly nonclassical photon statistics in parametric down-conversion,” Phys. Rev. A 73(3), 033814 (2006). [CrossRef]
51. C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44(5), R2775–R2778 (1991). [CrossRef]
52. J. Peřina Jr., V. Michálek, and O. Haderka, “Non-classicality of optical fields as observed in photocount and photon-number distributions,” Opt. Express 28(22), 32620–32631 (2020). [CrossRef]
53. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 1 (McGraw–Hill, 1953).
54. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Statist. Soc. B 39(1), 1–38 (1977).
55. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic Press, 2000).
