Figure 1

An outline of our experimental setup. Sources $S_1$ and $S_2$ emit pairs of polarization-entangled photons. The entangled pairs are emitted into spatial modes 1 and 3, and 2 and 4. One photon from each pair is directed into location $A$ and the other into location $B$. At the two locations photons impinge on beam splitters and are then detected by photodetectors. There are four possible outcomes in this experiment: coalescence at $A$ and coalescence at $B$, coalescence at $A$ and anticoalescence at $B$, anticoalescence at $A$ and coalescence at $B$, anticoalescence at $A$ and anticoalescence at $B$. The probabilities associated with the four outcomes are, respectively, $p_{cc},p_{ca},p_{ac},p_{aa}$ (subscript $c$ stands for coalescence and $a$ for anticoalescence). In terms of these probabilities the entropic inequalities (1) are written as $p_{ca}\ge p_{aa}$, $p_{ac}\ge p_{aa}$.Reuse & Permissions

Figure 2

Experimental setup. Two pairs of polarization-entangled photons are generated in two separate 3 mm-thick BBO crystals via type II SPDC. The crystals are pumped by two beams, each consisting of a train of femtosecond laser pulses (160 fs) with the central wavelength of 415 nm and a repetition rate of 76 MHz. The beams are displaced by a few micrometers using a trombone and the subwavelength path difference (phase shift $\varphi$) is introduced by a tilted 1 mm-thick quartz plate inserted into one of the pump beams. The two pump beams are obtained by splitting the main beam (average power 470 mW), which is generated by second harmonic generation (SHG) of the output of a Coherent MIRA-VERDI laser. Entangled photons emerging from the BBO crystals enter interferometers which provide high fidelity polarization-entangled states in an ultrafast regime [13]. Subsequently, the photons are sent to the beam splitters ${\mathrm{BS}}_1$ and ${\mathrm{BS}}_2$, so that each beam splitter receives two photons; one photon from each entangled pair. The photons are coupled in single-mode (SM) fibers [14] and detected by the photodetector (Perkin-Elmer SPCM-AQR-14). Once the correct position of the two beam splitters is established the fourfold coincidence interference patterns are measured. Coincidence counts $D_1\otimes D_2\otimes D_{3A}\otimes D_{4A}$, $D_1\otimes D_2\otimes D_{3A}\otimes D_{4B}$, $D_1\otimes D_2\otimes D_{3B}\otimes D_{4A}$ and $D_1\otimes D_2\otimes D_{3B}\otimes D_{4B}$ allow the estimation of the rate of anticoalescence-anticoalescence events, whereas $D_1\otimes D_2\otimes D_{3A}\otimes D_{3B}$ and $D_1\otimes D_2\otimes D_{4A}\otimes D_{4B}$ give the anticoalescence-coalescence rate.Reuse & Permissions

Figure 3

Possible emissions leading to four-photons coincidences. Parametric down-conversion is not an ideal source of entangled photons. It generates a superposition of vacuum, two-entangled photons, four-entangled photons, etc., The central diagram shows the desired emission of two independent entangled pairs—one by source $S_1$ and one by source $S_2$. The top and the bottom diagrams show unwelcome emissions of four photons by one of the two sources. Although the three cases are equally likely, we can eliminate contributions of unwelcome emissions by a judicious choice of the phase difference between the beams that coherently pump the two BBO crystals. We call it “phase marking.” Symmetric superposition of the unwelcome emissions, obtained for $\varphi =0$, does not contribute to the coalescence-anticoalescence events, whereas the antisymmetric superposition, obtained for $\varphi =\pi /2$ does not affect the anticoalescence-anticoalescence outcomes.Reuse & Permissions

Figure 4

Four-photon coincidences. The plot shows the four-photon interferences as a function of the pump displacement which is proportional to the relative phase shift $\varphi$ of the two pumps. The probability of coalescence and anticoalescence for the ideal singlet state case can be extracted from the coincidence counts at the minima of interference curves using (8). The minima of the curves correspond to the required values of $p_{ac}$ and $p_{aa}$. The experimental data show that $p_{aa}>p_{ac}$, which violates the entropic inequalities (1). The results for the curve anticoalescence-coalescence are normalized by a factor of 2 to take into account the fact that half of the coincidences are lost.Reuse & Permissions