Nonclassicality and entanglement criteria for bipartite optical fields characterized by quadratic detectors

Numerous inequalities involving moments of integrated intensities and revealing nonclassicality and entanglement in bipartite optical fields are derived using the majorization theory, nonnegative polynomials, the matrix approach, and the Cauchy-Schwarz inequality. Different approaches for deriving these inequalities are compared. Using the experimental photocount histogram generated by a weak noisy twin beam monitored by a photon-number-resolving intensified CCD camera, the performance of the derived inequalities is compared. A basic set of 10 inequalities suitable for monitoring entanglement of a twin beam is suggested. Inequalities involving moments of photocounts (photon numbers) as well as some containing directly the elements of photocount (photon-number) distributions are also discussed as a tool for revealing nonclassicality.

Figure 1

Scheme of the experimental setup: A twin beam originating in a nonlinear crystal (BBO) pumped by an ultrashort pulse generates a weak twin beam. The signal field and the idler field (after reflection on the mirror; HR) are filtered with a bandpass interference filter (IF) and then detected by an iCCD camera. The pump-beam intensity is actively stabilized with feedback provided by the detector (D).

Figure 2

(a) Experimental photocount histogram $f_{\mathrm{si}}(c_{\mathrm{s}},c_{\mathrm{i}})$ and reconstructed photon-number distributions $p_{\mathrm{si}}(n_{\mathrm{s}},n_{\mathrm{i}})$ obtained by (b) maximum-likelihood and (c) calibration methods.

Figure 3

Topo graphs of regularized quasidistributions $P_{\mathrm{si}}(W_{\mathrm{s}},W_{\mathrm{i}})$ of integrated intensities derived from (a) the experimental photocount histogram $f_{\mathrm{si}}$ (via its multi-mode Gaussian fit) for the ordering parameter $s=1$, (b) the photon-number distribution $p_{\mathrm{si}}$ reconstructed by the expectation-maximization approach (via the decomposition into Laguerre polynomials) for $s=0$, and (c) the photon-number distribution $p_{\mathrm{si}}$ reconstructed by the calibration method (via its multimode Gaussian fit) for $s=0.1$. In (a), the maximum of $P_{\mathrm{si}}$ within the white area equals $3.6\times {10}^{-2}$; in (c), $2.7\times {10}^{-2}$. When determining $P_{\mathrm{si}}$ in (a) and (c), one effective mode comprising the whole signal (idler) beam was assumed [3, 16, 21, 25].

Figure 4

(a) Normalized global nonclassicality criteria $\tilde{E}$ defined in Eqs (20, 21, 22, 23, 24, 25, 26, 27, 28, 29) and (b) the corresponding nonclassicality depths $\tau$. Values determined from the experimental photocount histogram are plotted with red stars; those appropriate for the reconstructed photon-number distribution reached by the maximum-likelihood (calibration) method by green triangles (solid blue curve). Some error bars are smaller than the symbols used.

Figure 5

Normalized global nonclassicality criteria $\tilde{E}$ defined in Eqs (30, 31, 32), (65), and (66). For description, see the caption to Fig. 4.

Figure 6

(a) Normalized global nonclassicality criteria $\tilde{D}$ defined in Eqs (47, 48, 49) and (b) the corresponding nonclassicality depths $\tau$. For description, see the caption to Fig. 4.

Figure 7

(a) Normalized global nonclassicality criteria $\tilde{T}$ defined in Eqs (50, 51, 52, 53, 54, 55) and (b) the corresponding nonclassicality depths $\tau$. For description, see the caption to Fig. 4.

Figure 8

(a) Normalized global nonclassicality criteria $\tilde{M}, \tilde{C}$, and $\tilde{D}$ defined in Eqs (73, 74, 75, 76, 77), (80), and (82) and (b) the corresponding nonclassicality depths $\tau$. For description, see the caption to Fig. 4.

Figure 9

Normalized global nonclassicality criteria $\tilde{N}$ defined in Eqs (93, 94, 95, 96, 97, 98). For description, see the caption to Fig. 4. Normalization is done with respect to the corresponding quantities $N^{\mathrm{ref}}$ determined for the factorized distribution $P_{\mathrm{s}}\left(W_{\mathrm{s}}\right)P_{\mathrm{i}}\left(W_{\mathrm{i}}\right)$.

Figure 10

Normalized global nonclassicality criterion ${\tilde{F}}_{k^{\prime }{l}^{\prime }1}$ given in Eq. (90) for (a) the experimental photocount histogram $f_{\mathrm{si}}$ and $k^{\prime }{l}^{\prime }=kk$ (red stars), $(k+1)k$ (green stars), $(k+2)k$ (yellow stars), $k(k+1)$ (light-blue stars), and $k(k+2)$ (dark-blue stars); (b) the photon-number distribution $p_{\mathrm{si}}$ reconstructed by the maximum-likelihood approach and $k^{\prime }{l}^{\prime }=kk$ (red triangles), $(k+1)k$ (green triangles), and $k(k+1)$ (blue triangles); and (c) the photon-number distribution $p_{\mathrm{si}}$ reconstructed by the calibration method for $k^{\prime }{l}^{\prime }=kk$ (solid blue curve), where ${\tilde{F}}_{kl1}\equiv [(k+1)(k+2)p_{\mathrm{si}}(k+2,l)+(l+1)(l+2)p_{\mathrm{si}}(k,l+2)-2(k+1)(l+1)p_{\mathrm{si}}(k+1,l+1)]/[(k+1)(k+2)p_{\mathrm{s}}^{\mathrm{P}}(k+2)p_{\mathrm{i}}^{\mathrm{P}}\left(l\right)+(l+1)(l+2)p_{\mathrm{s}}^{\mathrm{P}}\left(k\right)p_{\mathrm{i}}^{\mathrm{P}}(l+2)-2(k+1)(l+1)p_{\mathrm{s}}^{\mathrm{P}}(k+1)p_{\mathrm{i}}^{\mathrm{P}}(l+1)]$ and $p_a^{\mathrm{P}}\left(n\right)$ is the Poissonian distribution, with the mean $\langle n_a\rangle$ normalized such that $p_a^{\mathrm{P}}\left(0\right)=1, a=\mathrm{s},\mathrm{i}$.