Compound Twin Beams Without the Need of Genuine Photon-Number-Resolving Detection

A scheme for building stronger multimode twin beams from a greater number of identical twin beams sufficiently weak so that single-photon-sensitive on-off detectors suffice for their detection is studied. The statistical properties of these compound twin beams involving nonclassicality are analyzed for intensities up to hundreds of photon pairs. Their properties are compared with those of genuine twin beams that require photon-number-resolving detectors for their experimental investigation. The use of such compound twin beams for the generation of sub-Poissonian light and for the measurement of absorption with sub-shot-noise precision is analyzed. A suitable theoretical model for compound twin beams is developed to interpret the experimental data.

Figure 1

Substitution scheme for a stronger multimode field with a photon-number distribution $p(n)$ detected by an $N$-pixel spatially multiplexed PNRD, giving a photocount histogram $f(c)$. The stronger field is replaced by $N$ identical weak fields with photon-number distributions $p_{\mathrm{av}}(n)$, detected by APDs that provide photocount histograms $f_{\mathrm{av}}(c)$. The photon-number distribution $p(n)$ and photocount histogram $f(c)$ are obtained from the photon-number distributions $p_{\mathrm{av}}(n)$ and photocount histograms $f_{\mathrm{av}}(c)$ by $N$-fold convolution. The elements of the detection matrix $T(c,n;N)$ give the probabilities of detecting $c$ photocounts out of $n$ impinging photons with an $N$-pixel detector.

Figure 2

(a) Experimental setup for generation of photon pairs in a nonlinear ${\mathrm{LiIO}}_3$ crystal by short optical pulses, and their detection by single-photon APDs $D_1$ and $D_2$; a frequency filter $F_1$ and lenses $L_{1,2,3}$ transform the beams used. More details are given in the text. (b) Signal-APD and idler-APD detection windows and their grouping into triplets, which give measurements on compound TWBs formed by three constituting TWBs.

Figure 3

(a) Mean number of photons $⟨n_i⟩$ (photocounts $⟨c_i⟩$) and (b) Fano factor $F_{n,i}$ ($F_{c,i_1}$) of the idler field, (c) noise-reduction parameter $R_n$ ($R_c$), and (d) covariance $C_n$ ($C_c$) of the compound TWBs, depending on the number $N$ of grouped detection windows. Isolated symbols with error bars [in (a),(c),(d) they are smaller than the plotted symbols] are drawn for the experimental photocount histograms (red asterisks) and the fields reconstructed by the ML approach (green triangles). The solid (dashed) curves with symbols originate from the model of compound (genuine) TWBs detected by APDs ($N$-pixel PNRDs). The black plain solid curves are drawn for the model of compound TWBs with correlations in intra-pump-pulse intensity fluctuations. In (a),(c),(d), the red solid and dashed curves nearly coincide. The horizontal plain dashed lines indicate the border between the classical and nonclassical regions ($F=1$, $R=1$).

Figure 4

(a),(b) Photon-number distributions $p_{si}(n_s,n_i)$ and (c),(d) corresponding quasidistributions $P_{si,s}(W_s,W_i)$ of integrated intensities for $N=1$ [(a), (c)] and $N=1000$ [(b), (d)]. In (c) [(d)], $s=0.5$ [$s=0$]. In (a),(c), the values in the white areas are greater than those explicitly indicated on the $z$ scale.

Figure 5

(a) Effective detection efficiencies ${\eta }^{\mathrm{eff}}$ obtained with (large symbols) and without (small symbols) subtraction of the detector dark counts, plotted as functions of the number $N$ of grouped detection windows. (b) Normalized photocount correlation functions $K$ and (c) averaged normalized photocount correlation functions $\overline{K}$ depending on the relative distance $\mathrm{\Delta }j$ of the detection windows; ${\overline{K}}_{\mathrm{\Delta }j}\equiv \sum _{\mathrm{\Delta }k\in ⟨\mathrm{\Delta }j-\delta j,\mathrm{\Delta }j+\delta j⟩}{K}_{\mathrm{\Delta }k}/(2\delta j+1)$. The experimental data are plotted as isolated symbols with error bars [in (a) they are smaller than the plotted symbols] for the signal (red asterisks) and idler (green triangles) detectors. The solid (dashed) curves with symbols in (a) originate from the model of compound (genuine) TWBs detected by APDs ($N$-pixel PNRDs); the curves nearly coincide. The black plain solid curves in (a) are drawn for the model of compound TWBs with correlations in intra-pump-pulse intensity fluctuations. The dashed horizontal lines in (b),(c) indicate neutral correlations $K=\overline{K}=0$; $\delta j=24$.

Figure 6

Nonclassicality depths for (a,c) the joint photocount (${\tau }_c$) and (b,d) the joint photon-number (${\tau }_n$) distributions depending on the number $N$ of grouped detection windows. The NCIs $E_{001}$ (red asterisks), $E_{101}$ (green triangles), $E_{111}$ (blue diamonds), and $E_{211}$ (magenta circles) are used in (a,b); the NCI $M_{1001}$ (asterisks) is shown in (c,d). The experimental data are plotted by isolated symbols; their error bars are smaller than the plotted symbols. The solid (dashed) curves originate from the model of compound (genuine) TWBs detected by APDs ($N$-pixel PNRDs); the solid and dashed curves nearly coincide.

Figure 7

(a) Mean number of idler photons $⟨n_{c,i}⟩$ (photocounts $⟨c_{c,i}⟩$), (b) Fano factor $F_{n_c,i}$ ($F_{c_c,i}$), (c) optimal conditioning signal photocount number $c_s$, and (d) its probability $p_{c_s}$ of detection. (e) [(f)] photocount- [photon]-number NCDs ${\tau }_{L,c_c,i}$ [${\tau }_{L,n_c,i}$] for NCIs $L_{11}^1$ (red asterisks), $L_{21}^1$ (green triangles), $L_{31}^1$ (blue diamonds), and $L_{41}^1$ (magenta circles) belonging to conditional idler fields $p_{c,i}^c$ generated after registering the $c_s$ optimal signal photocounts indicated in (c), depending on the number $N$ of grouped detection windows. In (a),(b),(d), isolated symbols with error bars (smaller than the plotted symbols) are drawn for the experimental photocount histograms (red asterisks) and the fields reconstructed by the ML approach (green triangles). The experimental data plotted in (e),(f) by isolated symbols have error bars smaller than the symbols used. The solid (dashed) curves arise from the model of compound (genuine) TWBs detected by APDs ($N$-pixel PNRDs). In (a),(b),(d),(e),(f), the solid and dashed curves nearly coincide. The horizontal plain dashed line in (b) indicates the nonclassicality border $F=1$.

Figure 8

Normalized relative error ${\delta }_r{c}_i$ of the idler photocount number for $N=200$ grouped detection windows depending on the number $N_M$ of measurement repetitions. The relative experimental errors of ${\delta }_r{c}_i$ plotted are given by $1/\sqrt{N_M}$.

Figure 9

(a) Mean numbers of idler photocounts $⟨c_i⟩$ and conditioned idler photocounts $⟨c_{c,i}⟩$; (b) the corresponding relative errors $\delta c_i$ and $\delta c_{c,i}$; (c) [(d)] normalized relative errors ${\delta }_r{c}_i$ [${\delta }_r{c}_s$] and ${\delta }_r{c}_{c,i}$ [${\delta }_r{c}_{c,s}$]; and (e) ratios $S_{c_s}$ and $S_{c_i}$ of normalized relative errors, depending on the number $N$ of grouped detection windows. The isolated symbols with error bars (smaller than the plotted symbols) originate from the experiment with signal [idler] photocounts (blue diamonds [red asterisks]) and signal [idler] conditioned photocounts (magenta circles [green triangles]). The solid (dashed) curves are given by the model of compound (genuine) TWBs detected by APDs ($N$-pixel PNDRs); $N_M=500$.